An Introduction to Frames and Riesz Bases - Christensen Ole

1 Frames in Finite-dimensional Inner Product Spaces
$1.3 Frames in $\mathbb{C}_n$$
1.5 Pseudo-inverses and the singular value decomposition
2 Infinite-dimensional Vector Spaces and Sequences
$3.3 Bases and biorthogonal systems in $\mathcal{H}$$
6.2 Riesz frames and near-Riesz bases
6.3 Frames containing a Riesz basis
6.4 A frame which does not contain a basis
9.2 Representations of the Gabor frame operator and duality
10.2 Discrete Gabor systems through sampling
$10.3 Gabor frames in $\mathbb{C}^L$$ 10.4 Shift-invariant systems
11.2 Sufficient and necessary conditions
12.4 Frames and multiresolution analysis
12.5 Exercises
14 Wavelet Frames via Extension Principles
14.8 Construction of pairs of dual wavelet frames
15.6 Perturbation of the Haar wavelet
15.7 Exercises
16 Approximation of the Inverse Frame Operator
Автор: Christensen Ole
Теги: mathematics
ISBN: 3-7643-4295-1
Год: 2002
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Текст
                    Ole Christensen
An Introduction to Frames and Riesz Bases
Birkhauser
Boston • Basel • Berlin
Applied and Numerical Harmonic Analysis
Published titles
JM Cooper: Introduction to Partial Differential Equations with MATLAB
(ISBN 0-8176-3967-5)
C.E. DAttellis and EM Fernandez-Berdaguer: Wavelet Theory and Harmonic Analysis in
Applied Sciences (ISBN 0-8176-3953-5)
H.G. Feichtinger and T. Strohmer: Gabor Analysis and Algorithms
(ISBN 0-8176-3959-4)
T.M. Peters, J.H.T. Bates, G.B. Pike, P. Munger, and J.C. Williams: Fourier Transforms and
Biomedical Engineering (ISBN 0-8176-3941 -1}
A.I. Saichev and W.A. Woyczynski: Distributions in the Physical and Engineering Sciences (ISBN 0-8176-3924-1)
R. Tolimierei and M. An: Time-Frequency Representations (ISBN 0-8176-3918-7)
G.T. Herman: Geometry of Digital Spaces (ISBN 0-8176-3897-0)
A. Prochazka, J. Uhlir, P.J.W. Rayner, and N.G. Kingsbury: Signal Analysis and Precfction (ISBN 0-8176-4042-8)
J. Ramanathan: Methods of Applied Fourier Analysis (ISBN 0-8176-3963-2)
A. Teolis: Computational Signal Processing with Wavelets (ISBN 0-8176-3909-8)
W.O. Bray and C.V. Stanojevic: Analysis of Divergence (ISBN 0-8176-4058-4)
G.T. Herman and A. Kuba: Discrete Tomography (ISBN 0-8176-4101-7)
J.J. Benedetto and P.J.S.G. Ferreira: Modem Sampling Theory
(ISBN 0-8176-4023-1)
A. Abbate, C.M. DeCusatis, and P.K. Das: Wavelets and Subbands
(ISBN 0-8176-4136-X)
L. Debnath: Wavelet Transforms and Time-Frequency Signal Analysis
(ISBN 0-8176-4104-1)
K. Grbchenig: Foundations of Time-Frequency Analysis (ISBN 0-8176-4022-3)
D.F. Walnut: An Introduction to Wavelet Analysis (ISBN 0-8176-3962-4)
O. Bratelli and P. Jorgensen: Wavelets through a Looking Glass (ISBN 0-8176-4280-3)
H. Feichtinger and T. Strohmer: Advances in Gabor Analysis (ISBN 0-8176-4239-0)
O. Christensen: An Introduction to Frames and Riesz Bases (ISBN 0-8176-4295-1)
Forthcoming Titles
L. Debnath: Wavelets and Signal Processing (ISBN 0-8176-4235-8)
E. Prestini: The Evolution of Applied Harmonic Analysis (ISBN 0-8176-4125-4)
G. Bi and Y.H. Zeng: Transforms and Fast Algonthms for Signal Analysis and Representations (ISBN 0-8176-4279-X)
Applied and Numerical Harmonic Analysis
Senes Editor
John J. Benedetto
Univers'ty of Maryland
Editorial Advisory Board
Ole Christensen
Technical University of Denmark
Department of Mathematics
DK-2800 Lyngby
Denmark
Library of Congress Cataloging-in-Publication Data
Christensen, Ole, 1966-
An introduction to frames and Riesz bases / Ole Christensen.
p. cm, - (Applied and numerical harmonic analysis)
Includes bibliographical references and index.
ISBN 0-8176-4295-1 (alk. paper) - ISBN 3-7643-4295-1 (alk. paper)
1, Frames (Vector analysis) 2. Bases (Linear topological spaces) 3. Signal processing-Mathematics. I. Tide. II, Series.
QA433.C47 2002
515'.63-dc2I
2002034519 CIP
AMS Subject Classifications: 41-01,41-02,42-01,42-02,4205,42C40
Printed on acid-free paper. ©2003 Birkhauser Boston
Birkhauser
All rights reserved. This work may not be translated or copied in whole or in part without the written permission of the publisher (Birkhauser Boston, c/o Springer-Verlag New York, Inc., 175 Fifth Avenue, New York, NY 10010, USA), except for brief excerpts in connection with reviews or scholarly analysis. Use in connection with any form of information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed is forbidden.
The use of general descriptive names, trade names, trademarks, etc., in this publication, even if the former are not especially identified, is not to be taken as a sign that such names, as understood by the Trade Marks and Merchandise Marks Act, may accordingly be used freely by anyone.
ISBN 0-8176-4295-1 SPIN 10875334
ISBN 3-7643-4295-1
Typeset by the author.
Printed in the United States of America.
987654321
Birkhauser Boston • Basel • Berlin
A member of BerielsmannSpringer Science+Business Media GmbH
Series Preface
The Applied and Numerical Harmonic Analysis (ANHA) book series aims to provide the engineering, mathematical, and scientific communities with significant developments in harmonic analysis, ranging from abstract harmonic analysis to basic applications. The title of the series reflects the importance of applications and numerical implementation, but richness and relevance of applications and implementation depend fundamentally on the structure and depth of theoretical underpinnings. Thus, from our point of view, the interleaving of theory and applications and their creative symbiotic evolution is axiomatic.
Harmonic analysis is a wellspring of ideas and applicability that has flourished, developed, and deepened over time within many disciplines and by means of creative cross-fertilization with diverse areas. The intricate and fundamental relationship between harmonic analysis and fields such as signal processing, partial differential equations (PDEs), and image processing is reflected in our state of the art ANHA series.
Our vision of modern harmonic analysis includes mathematical areas such as wavelet theory, Banach algebras, classical Fourier analysis, timefrequency analysis, and fractal geometry, as well as the diverse topics that impinge on them.
For example, wavelet theory can be considered an appropriate tool to deal with some basic problems in digital signal processing, speech and image processing, geophysics, pattern recognition, biomedical engineering, and turbulence. These areas implement the latest technology from sampling methods on surfaces to fast algorithms and computer vision methods. The underlying mathematics of wavelet theory depends not only on classical Fourier analysis, but also on ideas from abstract harmonic analysis, including von Neumann algebras and the affine group. This leads to a study of the Heisenberg group and its relationship to Gabor systems, and of the metaplectic group for a meaningful interaction of signal decomposition methods. The unifying influence of wavelet theory in the aforementioned topics illustrates the justification for providing a means for centralizing and disseminating information from the broader, but still focused, area of harmonic analysis. This will be a key role of ANHA, We intend to publish with the scope and interaction that such a host of issues demands.
Along with our commitment to publish mathematically significant works at the frontiers of harmonic analysis, we have a comparably strong commitment to publish major advances in the following applicable topics in
vi Series Preface
which harmonic analysis plays a substantial role:
Antenna theory Biomedical signal processing Digital signal processing
Fast algorithms Gabor theory and applications I mage processing Numerical partial differential equations
Prediction theory Radar applications
Sampling theory Spectral estimation Speech processing Time-frequency and time-scale analysis
Wavelet theory
The above point of view for the ANNA book series is inspired by the history of Fourier analysis itself, whose tentacles reach into so many fields.
In the last two centuries Fourier analysis has had a major impact on the development of mathematics, on the understanding of many engineering and scientific phenomena, and on the solution of some of the most important problems in mathematics and the sciences. Historically, Fourier series were developed in the analysis of some of the classical PDEs of mathematical physics; these series were used to solve such equations. In order to understand Fourier series and the kinds of solutions they could represent, some of the most basic notions of analysis were defined, e.g., the concept of “function.” Since the coefficients of Fourier series are integrals, it is no surprise that Riemann integrals were conceived to deal with uniqueness properties of trigonometric series. Cantor’s set theory was also developed because of such uniqueness questions.
A basic problem in Fourier analysis is to show how complicated phenomena, such as sound waves, can be described in terms of elementary harmonics. There are two aspects of this problem: first, to find, or even define properly, the harmonics or spectrum of a given phenomenon, e.g., the spectroscopy problem in optics; second, to determine which phenomena can be constructed from given classes of harmonics, as done, for example, by the mechanical synthesizers in tidal analysis.
Fourier analysis is also the natural setting for many other problems in engineering, mathematics, and the sciences. For example, Wiener’s Taube-rian theorem in Fourier analysis not only characterizes the behavior of the prime numbers, but also provides the proper notion of spectrum for phenomena such as white light; this latter process leads to the Fourier analysis associated with correlation functions in filtering and prediction problems, and these problems, in turn, deal naturally with Hardy spaces in the theory of complex variables.
Nowadays, some of the theory of PDEs has given way to the study of Fourier integral operators. Problems in antenna theory are studied in terms of unimodular trigonometric polynomials. Applications of Fourier analysis abound in signal processing, whether with the Fast Fourier transform (FFT), or filter design, or the adaptive modeling inherent in time
Series Preface vii
frequency-scale methods such as wavelet theory. The coherent states of mathematical physics are translated and modulated Fourier transforms, and these are used, in conjunction with the uncertainty principle, for dealing with signal reconstruction in communications theory. We are back to the raison d’etre of the ANHA series!
John J. Benedetto Series Editor University of Maryland College Park
To Khadija, Jakob; and K<
Contents
Preface	xvii
1	Frames in Finite-dimensional
Inner Product Spaces	1
1.1	Some basic facts about frames ........................... 2
1.2	Frame bounds and frame algorithms ...................... 10
1.3	Frames in Cn ........................................... 14
1.4	The discrete Fourier transform.......................... 19
1.5	Pseudo-inverses and the singular value decomposition . .	23
1.6	Finite-dimensional function spaces...................... 28
1.7	Exercises............................................... 32
2	Infinite-dimensional	Vector Spaces
and Sequences	35
2.1	Sequences............................................... 35
2.2	Banach spaces and Hilbert spaces........................ 38
2.3	L2(K) and ^2(N)......................................... 38
2.4	The Fourier transform .................................. 40
2.5	Operators on L2(K)...................................... 41
2.6	Exercises............................................... 42
3	Bases	45
3.1	Bases in Banach spaces.................................. 46
3.2	Bessel sequences	in Hilbert spaces...................... 50
xii
Contents
3.3	Bases and biort hogonal	systems in H..................... 54
3.4	Orthonormal bases........................................ 56
3.5	The Gram matrix.......................................... 60
3.6	Riesz bases.............................................. 63
3.7	Fourier series and Gabor bases........................... 69
3.8	Wavelet bases............................................ 72
3.9	Exercises................................................ 76
4	Bases and their Limitations	79
4.1	Gabor systems and the Balian-Low Theorem ................ 82
4.2	Bases and wavelets....................................... 83
4.3	General shortcomings..................................... 86
5	Frames in Hilbert Spaces	87
5.1	Frames and their properties.............................. 88
5.2	Frame sequences.......................................... 92
5.3	Frames and operators..................................... 93
5.4	Frames and bases......................................... 96
5.5	Characterization of frames.............................. 101
5.6	The dual frames......................................... Ill
5.7	Tight frames............................................ 115
5.8	Continuous frames ...................................... 115
5.9	Frames and signal processing............................ 117
5.10	Exercises.............................................. 119
6	Frames versus Riesz Bases	123
6.1	Conditions for a frame being a Riesz basis.............. 123
6.2	Riesz frames and near-Riesz bases....................... 126
6.3	Frames containing a Riesz basis......................... 126
6.4	A frame which does not contain a basis.................. 128
6.5	A moment problem........................................ 134
6.6	Exercises............................................... 136
7	Frames of Translates	137
7.1	Sequences in ........................................... 138
7.2	Frames of translates.................................... 140
7.3	Frames of integer-translates............................ 147
7.4	Irregular frames of translates.......................... 153
7.5	The sampling problem.................................... 156
7.6	Frames of exponentials.................................. 157
7.7	Exercises............................................... 163
8	Gabor Frames in L2(K)	167
8.1	Continuous representations.............................. 169
8.2	Gabor frames............................................ 171
Contents xiii
8.3	Necessary conditions.................................... 174
8.4	Sufficient conditions................................... 176
8.5	The Wiener space W...................................... 187
8.6	Special functions ...................................... 190
8.7	General shift-invariant systems......................... 192
8.8	Exercises............................................... 198
9	Selected Topics on Gabor Frames	201
9.1	Popular Gabor conditions ............................... 202
9.2	Representations of the Gabor frame operator and duality 204
9.3	The duals of a Gabor frame.............................. 208
9.4	The Zak transform....................................... 215
9.5	Tight Gabor frames...................................... 219
9.6	The lattice parameters.................................. 222
9.7	Irregular Gabor systems................................. 226
9.8	Applications of Gabor frames ........................... 230
9.9	Wilson bases............................................ 232
9.10	Exercises............................................... 233
10	Gabor Frames in ^2(Z)	235
10.1	Translation and modulation on ^2(Z)..................... 235
10.2	Discrete Gabor systems through sampling................. 236
10.3	Gabor frames in CL...................................... 244
10.4	Shift-invariant systems................................. 245
10.5	Frames in ^2(Z) and filter banks........................ 246
10.6	Exercises............................................... 248
11	General Wavelet Frames	249
11.1	The continuous wavelet transform........................ 251
11.2	Sufficient and necessary conditions .................... 253
11.3	Irregular wavelet frames ............................... 267
11.4	Oversampling of wavelet frames.......................... 270
11.5	Exercises............................................... 271
12	Dyadic Wavelet Frames	273
12.1	Wavelet frames and their duals ......................... 274
12.2	Tight wavelet frames.................................... 277
12.3	Wavelet frame sets...................................... 278
12.4	Frames and multiresolution analysis..................... 281
12.5	Exercises............................................... 281
13	Frame Multiresolution Analysis	283
13.1	Frame multiresolution analysis.......................... 284
13.2	Sufficient conditions................................... 286
13.3	Relaxing the conditions................................. 290
13.4	Construction of frames................................. 292
13.5	Frames with two generators............................. 308
13.6	Some limitations....................................... 310
13.7	Exercises.............................................. 311
14	Wavelet Frames via Extension Principles	313
14.1	The general setup...................................... 314
14.2	The unitary extension principle........................ 316
14.3	Applications to B-splines I............................ 323
14.4	The oblique extension principle........................ 328
14.5	Fewer generators ...................................... 331
14.6	Applications to B-splines II........................... 334
14.7	Approximation orders................................... 339
14.8	Construction of pairs of dual wavelet frames........... 341
14.9	Applications to B-splines III.......................... 344
14.10	Exercises.............................................. 345
15	Perturbation of Frames	347
15.1	A Paley-Wiener Theorem for frames...................... 348
15.2	Compact perturbation................................... 354
15.3	Perturbation of frame sequences........................ 356
15.4	Perturbation of Gabor frames........................... 358
15.5	Perturbation of wavelet frames......................... 361
15.6	Perturbation of the Haar wavelet....................... 362
15.7	Exercises.............................................. 362
16	Approximation of the Inverse Frame Operator	365
16.1	The first approach..................................... 365
16.2	A general method....................................... 369
16.3	Applications to Gabor frames........................... 376
16.4	Integer oversampled Gabor frames....................... 378
16.5	The finite section method ............................. 379
16.6	Exercises.............................................. 382
17	Expansions in Banach Spaces	383
17.1	Representations of locally compact	groups.............. 383
17.2	Feichtinger-Grochenig theory .......................... 388
17.3	Banach frames.......................................... 394
17.4	p-frames............................................... 397
17.5	Gabor systems and wavelets in LP(R)	and related spaces 400
17.6	Exercises.............................................. 401
Appendix A	403
A.l	Normed vector spaces and inner product	spaces..........	403
A.2	Linear algebra......................................... 404
u>nwiii& xv
А.З	Integration............................................ 405
А.4	Some special normed vector spaces...................... 406
A.5	Operators on Banach spaces............................. 407
A.6	Operators on Hilbert spaces............................ 408
A.7	The pseudo-inverse....................................  410
A.8	Some special functions................................. 412
A.9	B-splines.............................................. 413
A.10	Notes................................................. 416
List of symbols	419
References	421
Index	437
Preface
Frames have fascinated me since day one. Every student in mathematics learns about bases in vector spaces, allowing one to represent each element in a convenient and unique way. One day in 1990, Henrik Stetkaer, who was my masters thesis advisor, showed me the definition of a frame and told me that a frame is some kind of “overcomplete basis”: one can also represent each element in the vector space via a frame, but the representation might not be unique. I was really surprised: how come that the question in e.g., linear algebra always was how to extract a basis from an overcomplete set, and one never got the idea that overcompleteness by itself could be useful?
A search on Mathematical Reviews or Zentralblatt shows only a few titles of books or articles concerning frames published before 1991; among those we mention the original paper by Duffin and Schaeffer [121], the excellent book by Young [279], and the important papers by Daubechies, Grossmann and Meyer [108], Daubechies [105], and Heil and Walnut [172]. Now, just ten years later, hundreds of papers have the word frame in the title, and perhaps a thousand discuss one or more results. Today, no single book can treat all the important and interesting results that have been published.
The aim of this book is to present parts of the modern theory for bases and frames in Hilbert spaces in a way that the material can be used in a graduate course, as well as by professional readers. For use in a graduate course, a number of exercises is included; they appear at the end of each chapter. The number of exercises give a hint of the level of the chapter: there are many exercises in the introductory chapters, but only few in the advanced chapters. In the same spirit, almost all results in the introductory chapters appear with full proofs; in the advanced chapters several results
xviii Preface
are presented without proofs. We believe it is more useful to state a large number of results in a common framework than to see detailed proofs of significantly fewer statements; this feature also makes the book useful as a reference.
The content can be split naturally into three parts: Chapters 1-6 describe the theory on an abstract level, Chapters 7-14 describe explicit constructions in L2-spaces, and Chapters 15-17 deal with selected research topics.
In Chapters 1-6 almost all results concern frames in general Hilbert spaces. The goal is an almost complete treatment of the known results for frames. For the explicit constructions in L2(—7г,7г) and L2(R), which appear in Chapters 7-14, the situation is different. For this part, I was forced to concentrate on selected parts of the theory. Since we are mainly interested in overcomplete systems, the theory presented in these chapters is part of a larger picture, and the reader will certainly benefit from knowledge of the background. Chapter 7 connects to the theory for nonharmonic Fourier series, cf. the book [279] by Young. Gabor frames arise naturally in the context of time-frequency analysis, and the book by Grochenig [153] will clarify the role of Chapters 8-9 in time-frequency analysis. Finally, the role of wavelets is highlighted in the classic book [106] by Daubechies, which also gives the motivation for the study of frames arising from multiscale methods in Chapters 13-14. Technically, we do not rely on any of these books (only at a few places will we refer to results from them without proof), but they put the results of frame studies in the right perspective. Chapters 7-14 are also influenced by the fact that the material is used in several areas of applied mathematics; the reader will observe that although this is a book about mathematics, those chapters concentrate on applicable ways to construct frames rather than on abstract characterizations.
Let us describe the chapters in more detail. Chapter 1 presents basic results in finite-dimensional vector spaces with an inner product. This enables a reader with a basic knowledge of linear algebra to understand the idea behind frames without the technical complications in infinite-dimensional spaces. Many of the topics from the rest of the book are presented here, so Chapter 1 can also serve as an introduction to the later chapters.
Chapter 2 collects some definitions and conventions concerning infinitedimensional vector spaces. Special attention is given to the Hilbert space L2(R) and operators hereon. We expect the reader to be familiar with this material; the chapter is too short to be considered as an introduction to the subject.
Chapter 3 describes the classical theory for bases in Hilbert spaces and Banach theory. The examples in this chapter are chosen so they lead naturally to the constructions in Chapters 7-14.
Chapter 4 highlights some of the limitations on the properties one can obtain from bases, and thus provides motivation for considering generalizations of bases.
Preface xix
Chapters 5-6 contain the core material about frames and Riesz bases. Chapter 5 is classical, but Chapter 6 contains several results published in the last five years. The interplay between frames and bases is discussed in detail in Chapter 6, and we also discuss frames that become bases when a certain number of elements are deleted.
Chapters 7-14 deal with frames having a special structure. A central part deals with various sufficient conditions for existence of those frames. The most fundamental frames, namely frames of exponentials in L2(—7Г, тг) and frames of translates in L2(R), are discussed in Chapter 7. If one wants to consider frames in L2(R), these frames easily lead to Gabor frames, which is the subject of Chapters 8-10. Wavelet frames are introduced in Chapter 11, and sufficient conditions to find them are given for arbitrary dilation parameter a > 1 and translation parameter b > 0. Some results concerning irregular wavelet frames are also presented there. Chapter 12 specializes to the important case a = 2, b = 1, which has attracted much attention during the past ten years. Constructions via multiscale methods are the focus in Chapters 13-14.
In Chapter 15, the question is stability of frames, i.e., whether a set of elements close to a frame is itself a frame. Since real-life measurements are never completely exact, this question is very important for applications.
Chapter 16 presents methods for the approximation of the inverse frame operator using finite subsets of the frame. Since every application of frame theory has to be performed in a finite-dimensional vector space, this topic is also of fundamental importance for applications.
Chapter 17 deals with extensions and generalizations of the material from the previous chapters. Expansions in Banach spaces and their relationship to frames in Hilbert spaces are discussed, as well as frames appearing via integrable group representations. The latter subject gives a unified description of the frames from Chapters 8-11.
Finally, an Appendix collects several basic results for easy reference. It also contains material on pseudo-inverse operators and splines which is not expected to be known in advance, and therefore is treated in more detail.
For the purpose of a graduate course, we mention that if students have a good background in functional analysis, they can skip Chapter 1 and parts of Chapters 2-3. Chapter 4 is important as motivation and Chapter 5 is also core material. But after covering these chapters, a course can continue in several ways. One possibility is to follow a theoretical track, and consider the relationship between frames and bases in more detail; this could be followed by a study of one of the three final chapters. Another possibility is to continue with constructions of exponential frames and Gabor frames, or wavelet frames. If wavelets are chosen as the subject, it is worth noticing that the four wavelet chapters are almost independent of each other.
This book presents frames and Riesz bases from the functional analytic point of view, and we concentrate on their mathematical aspects. However, as demonstrated by several papers by Daubechies and others, frames are
XX
Preface
very useful in several areas of applied mathematics, including signal processing and image processing. But this part of the story should be told by the people who are directly involved in it, and we will only sketch a few applications.
It is a pleasure to thank the many colleagues and students who helped in the process of writing this book. The starting point was seventy pages of notes, which were written jointly with Torben Klint Jensen, who was at that time a masters student. My original idea was to write a book concentrating on frames in general Hilbert spaces; I am very happy that Thomas Strohmer and an anonymous reviewer suggested that I further go into detail with wavelet and Gabor systems. Their ideas added more than a hundred pages to the book, and extended the scope considerably. Very useful suggestions for adding material were also given by Hans Feichtinger.
Alexander Lindner read a large part of the final manuscript and proposed many improvements. Elena Cordero, Niklas Grip, Per Christian Hansen, Reza Mahdavi, John Rassias, Henrik Stetkaer, and Diana Stoeva read parts of the book and helped to spot mistakes; I am very grateful to all of them.
I am thankful to the Department of Mathematics at the Technical University of Denmark for providing me with the excellent working conditions that made it possible to concentrate on the book for two semesters. In addition, a large part of the book was written during a stay at the research group NuHAG at the University of Vienna. It is a pleasure to thank the group leader, Hans Feichtinger, and the members of NuHAG for their support.
I am thankful to John Benedetto for inviting me to write this book, and I thank the staff at Birkhauser, especially Tom Grasso and Ann Kost ant, for their assistance and support. Thanks are also given to Elizabeth Loew from Texniques, who helped with Latex problems.
Finally, I acknowledge support from the WAVE-program, sponsored by the Danish Science Foundation.
Ole Christensen Kgs. Lyngby, Denmark September 2002
An Introduction to Frames and Riez Bases
1
Frames in Finite-dimensional Inner Product Spaces
In the study of vector spaces one of the most important concepts is that of a basis, allowing each element in the space to be written as a linear combination of the elements in the basis. However, the conditions to a basis are very restrictive - no linear dependence between the elements is possible and sometimes we even want the elements to be orthogonal with respect to an inner product. This makes it hard or even impossible to find bases satisfying extra conditions, and this is the reason that one might look for a more flexible tool.
Frames are such tools. A frame for a vector space equipped with an inner product also allows each element in the space to be written as a linear combination of the elements in the frame, but linear independence between the frame elements is not required. Intuitively, one can think about a frame as a basis to which one has added more elements. In this chapter we present frame theory in finite-dimensional vector spaces. While this restriction makes part of the theory trivial, it also makes the basic idea more transparent. Furthermore, our intention is to present the results in such a way as to give the right feeling about the infinite-dimensional setting as well. This also means that we sometimes use unusual words ii^ the finite-dimensional setting. For example, we will frequently use the word “operator” for a linear map.
There are other reasons for starting with a chapter on finite-dimensional frames. Every “real-life” application of frames has to be performed in a finite-dimensional vector space, so even if we want to apply results from the infinite-dimensional setting, the frames will have to be confined to a finite-dimensional space at some point.
2
1. Frames in Finite-dimensional Inner Product Spaces
Most of the chapter can be fully understood with an elementary knowledge of linear algebra. In order not to make the proofs too cumbersome, we will at a few places use some more advanced results, mainly about norms in vector spaces; exact references to the Appendix or Chapter 2 will always be given in this case.
This chapter is organized as follows. Section 1.1 contains the basic properties of frames. For example, it is proved that every set of vectors {fk}™=i in a vector space with an inner product is a frame for span-t/jJ^Lj. We prove the existence of coefficients minimizing the /?2-norm of the coefficients in a frame expansion and show how a frame for a subspace leads to a formula for the orthogonal projection onto the subspace. In Section 1.2 the role of frame bounds is highlighted. Then, in Section 1.3 and Section 1.4 we consider frames in Cn; in particular, we prove how we can obtain an overcomplete frame by a projection of a basis for a larger space. We also prove that the vectors {fkYk^i in a frame for Cn can be considered as the first n coordinates of some vectors in Cm constituting a basis for C771, and that the frame property for {fk}™^ is equivalent to certain properties for the m x n matrix having the vectors Д as rows. In Section 1.5 we prove that the canonical coefficients from the frame expansion arise naturally by considering the pseudo-inverse of the pre-frame operator, and we show how to find it in terms of the singular value decomposition. Finally, in Section 1.6 we connect frames in finite-dimensional vector spaces with the infinite-dimensional constructions which will appear in later chapters.
1.1 Some basic facts about frames
Let V be a finite-dimensional vector space, equipped with an inner product which we choose to be linear in the first entry. Recall that a sequence {e/JjL-! in V is a basis for V if the following two conditions are satisfied:
(i) V = span{efc}^=1;
(ii)	is linearly independent, i.e., if	= 0 for some scalar
coefficients , then q = 0 for all к = 1,..., m.
As a consequence of this definition, every / G V has a unique representation in terms of the elements in the basis, i.e., there exist unique scalar coefficients {c/JJJLj such that
m
f ~^скек.	(1.1)
1.1 Some basic facts about frames
3
If {e/JJ™-! is an orthonormal basis, i.e., a basis for which
/	\ _ X _ f1 if k - j
10 if к j,
then the coefficients {qare easy to find: taking the inner product of f in (1.1) with an arbitrary ej gives
771	771
(/.e,) = {^скек,е,} = ^ck{ek,ej') = Cj, k=l	k=l
so
7П
/ = £(/,efc)ejfe.	(1.2)
k=l
We now introduce frames; in Theorem 1.1.5 below we prove that a frame {AlfcLi also allows a representation like (1.1).
Definition 1.1.1 A countable family of elements {fk}kei in V is a frame for V if there exist constants A, В > 0 such that
А НЛ12<Е1(ЛЛ)12<в ll/ll2, v/ev.	(1.3)
kei
The numbers A, В are called frame bounds. They are not unique. The optimal lower frame bound is the supremum over all lower frame bounds, and the optimal upper frame bound is the infimum over all upper frame bounds. Note that the optimal frame bounds are actually frame bounds. The frame is normalized if ЦДЦ = 1, V/c £ I.
In a finite-dimensional vector space it is somehow artificial (though possible) to consider families {fk}kei having infinitely many elements. In this chapter we will only consider finite families {fk}™-i, m E N. With this restriction, Cauchy-Schwarz’ inequality shows that
771	771
El(WI2 < EllAII2 ll/ll2, v/gv, /c=l	/c=l
i.e., the upper frame condition is automatically satisfied. However, one can often find a better upper frame bound than	| |Д 112, and we will soon
see that good estimates for the frame bounds are important.
In order for the lower condition in (1.3) to be satisfied, it is necessary that span{/fc}jIL1 = V. This condition turns out to be sufficient; in fact, every finite sequence is a frame for its span:
Proposition 1.1.2 Let {fk}^=1 be a sequence in V. Then {fk}™^ is a frame for span{fk}1fL1.
4
1. Frames in Finite-dimensional Inner Product Spaces
Proof. We can assume that not all fk are zero. As we have seen, the upper frame condition is satisfied with В = ЦАЦ2. Now let
W-span^}^,
and consider the continuous mapping
m ф-.W^K := 52l</, A>|2. /с=1
The unit ball in W is compact, so we can find g G W with ||^|| = 1 such that
m	f m	'I
>l:=ElW0|2=inf El</,A>|2 : feW, ||/|| = O-k=i	U=i	)
It is clear that A > 0. Now given f 6 W, /	0, we have
771<	771	p
Ei</-a>i2 = EKjiTii’^i2 A IK- □
Corollary 1.1.3 A family of elements	in V is a frame for V if
and only if spanlfk}™-^ = V.
Corollary 1.1.3 shows that a frame might contain more elements than needed to be a basis. In particular, if {fk}™-i is a frame for V and {£k}fc=i is an arbitrary finite collection of vectors in V, then {fk}™=1 U	is
also a frame for V. A frame which is not a basis is said to be overcomplete or redundant.
Consider now a vector space V equipped with a frame {fk}™=i and define
a linear mapping
m
T-.^^V, T{ck}^ =	(i-4)
fc=l
T is usually called the pre-frame operator, or the synthesis operator. The adjoint operator is given by
т*/ = {</,/0}Г=1.	П-5)
and is called the analysis operator. By composing T with its adjoint T*, we obtain the frame operator
m
S-	.V^V, Sf = TT*f = ^(f,fk}fk.	(1.6)
/c=l
Note that in terms of the frame operator,
m
<S/J) = Eltf--K’/е V;	(1-7)
k—1
1.1 Some basic facts about frames
5
the lower frame condition can thus be considered as some kind of “lower bound” on the frame operator.
A frame {A}E=i is tight if we can choose A = В in the definition, i.e., if
m
ЕКЛА)12=Л|И12, v/GV.	(1.8)
k=l
For a tight frame, the exact value A in (1.8) is simply called the frame bound. We note that (1.7) combined with Lemma A.6.6 immediately leads to a representation of f G V in terms of the frame elements:
Proposition 1.1.4 Assume that {fk}™-i is a tight frame for V with frame bound A. Then S = Al (here I is the identity operator on V), and
- m
(1.9)
A
An interpretation of (1.9) is that if {fk}£ЕХ is a tight frame and we want to express f E Vasa linear combination f = ckfk, we can simply define 9k — j{fk and take ck = {f,9k)- Formula (1.9) is similar to the representation (1.2) via an orthonormal basis: the only difference is the factor 1/A in (1.9). For general frames we now prove that we still have a representation of each f G V of the form f = Y^k=i{fi9k)fk for an appropriate choice of {9к}™~1- The obtained theorem is one of the most important results about frames, and (1.10) is called the frame decomposition:
Theorem 1.1.5 Let {fk}™-i be a frame for V with frame operator S. Then
(i) S is invertible and self-adjoint.
(ii) Every f EV can be represented as
m	m
f = YSf’S-1fk)fk = '£{f,fk)S-1fk.	(1.10)
fc=l	k=l
(in) Iff G V also has the representation f = ckfk for some scalar coefficients	then
mm	m
E ы2 = E к/-5-1 ля2+E ic* - </> 5-1a>i2-Л'=1	k=l	k=\
Proof. Since S = TT*, it is clear that S is self-adjoint. We now prove that S is injective. Let f EV and assume that Sf = Q. Then
m
o = (S/,/) = El(AA)l2’
6
1. Frames in Finite-dimensional Inner Product Spaces
implying by the frame condition that f = 0. That S is injective actually implies that S is surjective, but let us give a direct proof. The frame condition implies by Corollary 1.1.3 that span{/fc}JIL1 = V, so the pre-frame operator T is surjective. Given f 6 V we can therefore find g G V such that Tg = f\ we can choose g 6	= Нт* , so it follows that 1Zs = Utt* — V.
Thus S is surjective, as claimed. Each f EV has the representation
f = SS-'f
= TT'S-'f m
k=l
using that S is self-adjoint, we arrive at
m
f =
k—1
The second representation in (1.10) is obtained in the same way, using that f = S~1Sf. For the proof of (iii), suppose that f = ckfk- We can write
By the choice of {cfc}b=i we have
m
- (Л5-1 fk}) fk = 0,
i.e., {cfc}™=1 - {(/,S-1/k)}r=i £	= K^.; since
{(/,5-7^}?=! = {(5-7, e KT.,
we obtain (iii).	□
Every frame in a finite-dimensional space contains a subfamily which is a basis (Exercise 1.1). If {fk}™=i is a frame but not a basis, there exist non-zero sequences	such that <4 fk = 0- Therefore f E V
can be written
m	m
f = '£<f,S-1fk)fk + '£dkfk
fc=l	/c=l
m
= Е«Д5-7^+4)Л, fc=l
showing that f has many representations as superpositions of the frame elements. Theorem 1.1.5 shows that the coefficients {{fjS~1fk)}^==1 have minimal ^2-norm among all sequences {q}^=1 for which f = ^™-ickfk-
1.1 Some basic facts about frames
7
The numbers
(Л^М fc = l,...,m
are called frame coefficients. Note that because S : V -» V is bijective, the sequence {S~1fk}kL1 is also a frame by Corollary 1.1.3; it is called the canonical dual of {fk}™=i-
Theorem 1.1.5 gives some special information in case	is a basis:
Corollary 1.1.6 Assume that	is a basis for V. Then there exists
a unique family {<7/c}/2=i in V suc^ that m
f = ^{f,9k)fk^feV.	(1.11)
/c=l
In terms of the frame operator, {g/J/TLi = {5-1 fk}™-i- Furthermore {fj,9k} = $j,k-
Proof. The existence of a family	satisfying (1.11) follows from
Theorem 1.1.5; we leave the proof of the uniqueness to the reader. Applying (1.11) on a fixed element fj and using that {fk}™-i is a basis, we obtain that {fj, gk) = 5j,k for all к = 1,2, • • • , m.	□
We can give an intuitive explanation of why frames are important in signal transmission. A more detailed argument is given in [105]. Let us assume that we want to transmit the signal f belonging to a vector space V from a transmitter A to a receiver 7Z. If both A and 7Z have knowledge of a frame {fk}™^ for V, this can be done if A transmits the frame coefficients {(/, S~1fk)}^==1; based on knowledge of these numbers, the receiver 7Z can reconstruct the signal f using the frame decomposition. Now assume that 7Z receives a noisy signal, meaning a perturbation {(/,	+ q}/2=i
the correct frame coefficients. Based on the received coefficients, IZ will claim that the transmitted signal was
m	m	m	m
E «/, s-1 fk)+Ck) fk = е<л s-1 fk)fk + e tkh = f+E k—1	fc=l	fc=l	fe=l
this differs from the correct signal f by the noise Ckfk- If {A}^=i Is overcomplete, the pre-frame operator T{ck}™=1 = 52/Xi has a non" trivial kernel, implying that parts of the noise contribution might add up to zero and cancel. This will never happen if	is an orthonormal
basis? In that case || ckfk112 = SJXi Iе* P, so eac^ n°ise contribution will make the reconstruction worse.
We have already seen that, for given f G V, the frame coefficients {(/,have minimal £2-norm among all sequences {c/JJE^ for which f = Y^k~\ckfk‘ We can also choose to minimize the norm in other spaces than €2; we now show the existence of coefficients minimizing the ^-norm.
8
1. Frames in Finite-dimensional Inner Product Spaces
Theorem 1.1.7 Let {A}^=1 be a frame for a finite-dimensional vector space V. Given f 6 V, there exist coefficients	£ C771 such that
f - dkfk, and
m	Cm	m	'j
£|dfc|=inf £|cfc| : / = 52^/4-	(1.12)
/c=l	lfc=l	k=l	J
Proof. Fix f e V. It is clear that we can choose a set of coefficients R/JfcLi such that f = ckfk\ let r :=	|q|. Since we want to
minimize the ^-norm of the coefficients, it is also clear that we can now restrict our search for a minimizer to sequences {dk}™=1 belonging to the compact set
M '•={№ еГ : \dk\<r, fc = l,...,m}.
Now the result follows from the fact that the set
m
{4<=i e M | f = ^dkfk k=l
is compact and that the function ф : C77 —> R, Ф№к}™-1 •= SkLi l^fcl Is continuous.	□
There are some important differences between Theorem 1.1.5 and Theorem 1.1.7. In Theorem 1.1.5 we find the sequence minimizing the ^2-norm of the coefficients in the expansion of f explicitly; it is unique, and it depends linearly on f. On the other hand, Theorem 1.1.7 only gives the existence of an ^-minimizer, and it might not be unique (Exercise 1.4). Even if the minimizer is unique, it might not depend linearly on f (Exercise 1.5). An algorithm to find an ^-minimizer {dk}™--^ can be found in [64].
As we have seen in Proposition 1.1.2, every finite set of vectors {fk}™=i is a frame for its span. If spanf/^}^! V, the frame decomposition associated with {Д}^=1 gives a convenient expression for the orthogonal projection onto spanlJ/J^Lp
Theorem 1.1.8 Let {A}£=i be a frame for a subspace W of the vector space V. Then the orthogonal projection of V onto W is given by
m
Pf = ^{f,S-1fk)fk.	(1.13)
k=l
Proof. It is enough to prove that if we define P by (1.13), then
Pf = f for f e W and Pf = 0 for f e W\
The first equation follows by Theorem 1.1.5, and the second by the fact that the range of S'"1 equals W because S is a bijection on W.	□
1.1 Some basic facts about frames
9
Example 1.1.9 Let {e/JjLj be an orthonormal basis for a two-dimensional vector space V with inner product. Let
/1 = ei, /2 = ei - 62, /3 = 61 + 62-
Then {fk}k=i is a frame for V. Using the definition of the frame operator,
3
k=l
we obtain that
Sei = 6i + 6i — 62 + 6i + 62 = 3d, Se? = — (ei — 62) + 6i + 62 — 2e2.
Thus
S-1ei = Li, S-1e2 = L2. О	Z
Therefore the canonical dual frame is
{s~7oU = {U, к - к, |ei + L2j. OO z о z
Via Theorem 1.1.5, the representation of / G V in terms of the frame is given by
3
/ = £(/,s-7k>A fc=l
=	+ (/, Li - |e2)(ei - e2) + {f, je, + |e2)(ei +e2). □
о	о z	о z
Let us for a moment consider an orthonormal basis {ek}£=1 for V. It is clear that by adding a finite collection of vectors to	we obtain a
frame for V. Also, if we perturb the vectors	slightly, we still have
a basis, but in general not an orthonormal basis. More precisely, if {дк}%=1 is a family of vectors in V and
/ n	\ 1/2
El|e,-^||2	<1,
\/c=l	/
then also {gk}k=i Is a basis for V. In fact, given a scalar sequence {ca:}J==1, the opposite triangle inequality followed by Cauchy—Schwarz’ inequality
10
1. Frames in Finite-dimensional Inner Product Spaces
gives that
n
ck9k k=l
n
^с^дк-ек) k=i
n ^2Ckek k=l
\ 1/2 / n \ !/2 52l|efc-rf I (5LlCfc|2 ьк—1	/ \fc=l /
= (1-H)
1/2
This shows that {gk}k=i is linearly independent, and since dimV = n, we conclude that	is a basis. We return to more general perturbation
results for frames in Chapter 15.	□
1.2 Frame bounds and frame algorithms
The speed of convergence in numerical procedures involving a strictly positive definite matrix depends heavily on the condition number of the matrix, which is defined as the ratio between the largest eigenvalue and the smallest eigenvalue. In case of the frame operator, these eigenvalues correspond to the optimal frame bounds:
Theorem 1.2.1 Let	be a frame for V. Then the following holds:
(i) The optimal lower frame bound is the smallest eigenvalue for S, and the optimal upper frame bound is the largest eigenvalue.
(ii) Let {}JJ=1 denote the eigenvalues for S; each eigenvalue appears in the list corresponding to its algebraic multiplicity. Then
n	m
k=l
(Hi) Assume that V has dimension n. If {fk}™^ is tight and ||A|| — 1 for all к, then the frame bound is A = m/n.
Proof. Assume that {А}^ is a frame for V. Since the frame operator S : V -> V is self-adjoint, Theorem A.2.1 shows that V has an orthonormal basis consisting of eigenvectors	for S. Denote the corresponding
eigenvalues by {А/с}£_1. Given f G V, we can write f = Y?k=i{f>ek)ek-Then
n	n
Sf = ^,ек)Зек = ^Ы/,ек)ек, k=l	k=l
1.2 Frame bounds and frame algorithms
11
and m	n
ЕК/,Л)|2 = (5/,Л = ^Xk\{f,ek)\2. k=l	k=l
Therefore
m Aminll/ll2 < El</, A>|2 < Amax||/||2. /с —1
So Amin is a lower frame bound, and Amax is an upper frame bound. That they are the optimal frame bounds follows by taking f to be an eigenvector corresponding to Amin (respectively Amax).
For the proof of (ii), we have
E = E = E(5efc,efc) k=l k=l n m
= EEkw,)i2. k=l1=1
Interchanging the sums and using that {e/JJEj is an orthonormal basis for V finally gives (ii). For the proof of (iii), the assumptions imply that the set of eigenvalues {AfcJJLj consists of the frame bound A repeated n times; thus the result follows from (ii).	□
Corollary 1.2.2 Let {fk}™-i be a frame for V. Then the condition number for the frame operator is equal to the ratio between the optimal upper frame bound and the optimal lower frame bound.
If we want to find an element f EV based on knowledge of the coefficients {(/, fk)}™-i we can use Theorem 1.1.5:
m
/c=l
However, in order for this formula to be useful we need to invert the frame operator, which can be complicated if the dimension of V is large. Another option is to use an algorithm to obtain approximations of f. A classical algorithm is known as the frame algorithm:
Lemma 1.2.3 Let {fk}™-i be a frame for V with frame bounds A,B. Given f EV, define the sequence {дк}&=о V by
2
9o=0, 9k = 9k-i +- gk-i), к >1.	(1.14)
A + Jd
Then
f В — A\k
n/-^ll< rT7 ll/ll-
\jd + A J
12
1. Frames in Finite-dimensional Inner Product Spaces
Proof. Let I denote the identity operator on V. Using (1.7),
<(/ ~	= IK" aTb Pv/ e v'
so via the frame condition,
<(/-	-ГГв 11/1,2 =
Similarly, p _ д	о
The two inequalities and (A.8) together give that
I-
A + B
B — A - В + A'
Using the definition of {#k}£l0,
2
f ~ 9k = f ~ 9k-i ~	- Qk-i)
and by repeating the argument,
f - 9k
/	2	\к
V-aTb8)
Thus, applying (A.6) and (A.7),

/	2	\k
V-aTb8)
2 к ^АТв8 "/-Poll
\B + A J ,,Л|
□
In particular, the vectors gk in (1.14) converge to f as к -> oo. The algorithm depends on the knowledge of some frame bounds, and the guaranteed speed of convergence also depends on them. If В is much larger than A (either because only bad estimates for the optimal bounds are known, or because the frame is far from being tight) the convergence might be slow.
1.2 Frame bounds and frame algorithms
13
It is natural to apply some of the known acceleration algorithms from linear algebra to obtain faster convergence. Grochenig showed in [152] how to apply the Chebyshev method and the conjugate gradient method. We begin with the Chebyshev method:
Theorem 1.2.4 Let	be a frame for V with frame bounds A,B,
and let
В-A VB-VA
P'~ B + A' a y/B + VA’
Given f EV, define the sequence {дк}^о V and corresponding numbers by
2
9o = 0, gi = у	Ai = 2,
Л + Jd
and for к >2,
1	/	2	\
^k =	= Afc I gk-1 - gk-2 4- — nS(f - 9k-i) + 9k-2-
1 -	\	ал- a	/
Then
е2стк
ll/-^l<rT^JI/ll-
The Chebyshev algorithm guarantees a faster convergence than the frame algorithm when В is much larger than A. Knowledge of some frame bounds is also needed in order to apply the Chebyshev algorithm. In contrast, the conjugate gradient algorithm described below works without knowledge of the frame bounds: only when we want to estimate the error \\f — дь\\ do we need them. Following Grochenig, we formulate the result using the norm
111/111 = </,S/)1/2, /ev.
We leave it to the reader to check that ||| • ||| is in fact a norm on V. Remember also that all norms on a finite-dimensional vector space are equivalent; that is, there exist constants Ci, C2 > 0 such that
Cdl/ll < lll/lll <6/211/11, v/ev
This means that an error estimate in the norm ||| • ||| can be transferred into an error estimate in the usual norm.
Theorem 1.2.5 Let {fk}™-i be a frame for V. Let f G V\{0} and define the vectors {gk}kLo, {rk}kL0, {PkjkL-т and numbers {Afc}£T0 by
go = 0, r0 = p0 = Sf, p-i = 0
14
1. Frames in Finite-dimensional Inner Product Spaces
and, for к >0,
\	= (rk,pk)
k {Pk,Spk}'
9k+i = 9k + Xkpk,
rk+i = rk - XkSpk, c„ (Spk,Spk)~ {Spk,Spk-i)
Pfc+1 ^Pk I Q \ Pk / Cj	\Pk —!•
\Pki$Pk) \Pk-l, SPk-1)
Then pk -> f as к -» oo. If we let A denote the smallest eigenvalue for S and В the largest eigenvalue and let a =	the speed of convergence
can be estimated by
III/-Ы1 <^*111/111.
In the expression for pk+i, the last term is interpreted as zero for к » 0.
1.3 Frames in Cn
The natural examples of finite-dimensional vector spaces are
Kn = {(ci,C2,...,Cn) I Ci e IR, 2 = 1,...,™} and
Cn = {(ci,c2, • •.,Cn) I Сг e C, 2 = 1,...,™}; the latter is equipped with the inner product n ({ck}fc=l, {^A:}fc-1) — Ckdk k=l and the associated norm
11ЬК=1Н = A £Ы2-
This corresponds to the definitions in IRn, except that complex conjugation and modulus is not needed in the real case. We will describe the theory for bases and frames in Cn, but easy modifications give the corresponding results in IRn. If, for example, {fk}™=i is & frame for Cn, then the 2m vectors consisting of the real parts, respectively the imaginary parts, of the frame vectors will be a frame for IRn (Exercise 1.6); in particular, if the vectors {fk}™-! have real coordinates, they constitute a frame for IRn. On the other hand a frame for is automatically a frame for Cn (Exercise 1.7).
1.3 Frames in C™
15
The canonical basis for Cn consists of the vectors	where 5 k
is the vector in C71 having 1 at the к-th entry and otherwise 0. We will consequently identify vectors in Cn with their representation in this basis.
From elementary linear algebra we know many equivalent conditions for a set of vectors to constitute a basis for Cn. Let us list the most important characterizations:
Theorem 1.3.1 Consider n vectors in Cn and write them as columns in an n x n matrix
A12
A22
Then the following are equivalent:
(i) The columns in A (i.e., the given vectors) constitute a basis for C*.
(ii) The rows in A constitute a basis for C1.
(Hi) The determinant of A is non-zero.
(iv)	A is invertible.
(v)	A defines an injective mapping on Cn.
(vi)	A defines a surjective mapping on Cn.
(vii)	The columns in A are linearly independent.
(viii)	A has rank equal to n.
Recall that the rank of a matrix E is defined as the dimension of its range Ke- We also remind the reader that any basis can be turned into an orthonormal basis by applying the Gram-Schmidt orthogonalization procedure.
We now turn to a discussion of frames for Cn. Note that we consequently identify operators V : Cn —> Cm with their matrix representations with respect to the canonical bases in Cn and C71.
In case {fk}™-i is a frame for Cn, the pre-frame operator T maps C771 onto Cn, and its matrix with respect to the canonical bases in Cn and Cm is
/	I	I	•	•	I
T =	Л	f2			fm
\	I	I	•	•	I
(1-15)
i.e., the n x m matrix having the vectors Д as columns.
16
1. Frames in Finite-dimensional Inner Product Spaces
Since m vectors can at most span an m-dimensional space, we necessarily have m > n when {fk}™-i is a frame for Cn, i.e., the matrix T has at least as many columns as rows.
Theorem 1.3.2 Let {A}fcLi be a frame for C1. Then the vectors fk can be considered as the first n coordinates of some vectors gk in C777 constituting a basis forCm. If {fk}™^ is a tight frame, then the vectors fk are the first n coordinates of some vectors gk in FF constituting an orthogonal basis for Cm.
Proof. Let {A}£=i be an arbitrary frame for Cn. Then m > n. Consider the mapping
Fz = {(x,fk}№=l.
F is the adjoint of the pre-frame operator T, and the matrix for F with respect to the canonical bases is the m x n matrix where the к-th row is the complex conjugate of fk, i.e.,
/	-	л	-	\
-	A	-
F =
\	~	fm	~	/
If Fx = 0, then 0 = ||Fz||2 =	|<z, A)|2. Since span{A}£Li = it
follows that x — 0, so F is an injective mapping. We can therefore extend F to a bijection F of Cm onto CF: for example, still letting	be
the canonical basis for Cm, let {<A}£Ln+1 be a basis for the orthogonal complement of 1Zf in C771 and extend F by the definition Fdk := фк, к = n 4- 1, n + 2,..., m. The matrix for F is an m x m matrix, whose first n columns are the columns from F:
/ -	7Г	-	I	I	•	I	\
= I ’	* I фп+l  фтп I •
\ -	7^	-	I	I	•	I	/
Since F is surjective, the columns span C771. The rank of the rows equals the rank of the columns, so also the rows in F span CF, and they are linearly independent. Thus, they constitute a basis for C771.
If {A}£zi is a tight frame for C71 with frame bound A and {A}£=i denotes the canonical basis for Cn, Proposition 1.1.4 shows that
{TT*6i,6j) is the j, l-th entry in the matrix representation for TT*, so this calculation shows that the n rows in the matrix representation (1.15) for T are orthogonal, considered as vectors in Cm. By adding m — n rows we
1.3 Frames in C1
17
can extend the matrix for T to an m x m matrix in which the rows are orthogonal. Therefore the columns are orthogonal.	□
Geometrically, Theorem 1.3.2 means that if {fk}™-! is a frame for O1, there exist vectors {hk }J£-i in C™-71 such that the columns in the matrix
/ I	I	•	•	I \
/1	/2	•	•	fm
I	I	*	•	I
hj	/^2	’		hm
\ \ \   \ J
(1-16)
constitute a basis for Cm.
For a given m x n matrix A the following proposition gives a condition for the rows constituting a frame for Cn.
Proposition 1.3.3 For an m x n matrix
/	Ац	•	♦	Aln
л=	'	;	;	*
\	•	•	Amn	J
the following are equivalent:
(i) There exists a constant A > 0 such that n
АЕы2<ЦЛ{Ск}^||2, Ж}ыеС".
(ii) The columns in A constitute a basis for their span in Cm. (iii) The rows in A constitute a frame for C1.
Proof. Denote the columns in A by £1,... ,<7™; they are vectors in C™. By definition, (i) means that for all	G Cn,
<
52 Ck9k , k=l
(1-17)
which is equivalent to {gk}™-! being a basis for its span in (use an argument such as in the proof of Proposition 1.1.2). On the other hand, denoting the rows in A by /1,..., fm, (i) can also be written as
(A,
/сЛ
C2
\Cn/
2
, v{Cfc}Li
e C*,
which is equivalent to (iii).
□
18
1. Frames in Finite-dimensional Inner Product Spaces
As an illustration of Proposition 1.3.3, consider the matrix
/	1	°	\
A =	0	1	;
\	1	0	/
it is clear that the rows	constitute a frame for C2.
/ i \ / 0 \
The columns I 0 I , I 1 constitute a basis for their span in C3, but \ 1 / \ 0 /
the span is only a two-dimensional subspace of C3.
As an immediate consequence of the proof of Proposition 1.3.3 we have
the following useful fact:
Corollary 1.3.4 Let A be an m x n matrix. Denote the columns by gi,... ,gn and the rows by fi,..., fm. Given A,B > 0, the vectors {fk}™-! constitute a frame for Cn with bounds A, В if and only if
A±\c^< k=l
n
У ck9k
k=l
<B^\ck\\ v{Q}^ee.
Example 1.3.5 Consider the vectors
(1.18)
A =
The reader can check that the columns	are orthogonal in C5 and
all have length w |. Therefore
з	з
Y^Ckgk = |^2Ы2
for all Ci, C2, C3 G C. By Corollary 1.3.4 we conclude that the vectors defined by (1.18) constitute a tight frame for C3 with frame bound The frame is normalized.	□
1.4 The discrete Fourier transform
19
For later use we state a special case of Corollary 1.3.4 (Exercise 1.8):
Corollary 1.3.6 Let A be an m x n matrix. Then the following are equivalent.
(i) A* A = I, the n x n identity matrix.
(ii) The columns gi,... ,gn in A constitute an orthonormal system in C771.
(Hi) The rows Д,..., fm in A constitute a tight frame for Cn with frame bound equal to 1.
1.4 The discrete Fourier transform
When working with frames and bases in Cn one has to be particularly careful with the meaning of the notation. For example, we have used Д and gk to denote vectors in Cn, while q in general is the /с-th coordinate of a sequence	G C1, i.e., q is a scalar. In order to avoid confusion
we will change the notation slightly in this section. The key to the new notation is the observation that to have a sequence in Cn is equivalent to having a function
f :{1,.
the j-th entry in the sequence corresponds to the j-th function value f(j).
Our purpose is to consider a special orthonormal basis for Cn. Given f G Cn we denote the coordinates of f with respect to the canonical orthonormal basis	by	For к = 1,... , n we define vectors
ек G C1 by
ek(j) = -he2-b-i)^, j =	(1.19)
x/n
that is
(1.20)
Theorem 1.4.1 The vectors {е/с}^ defined by (1.19) constitute an orthonormal basis for Cn.
Proof. Since {e/c are n vectors in an n-dimensional vector space, it is enough to prove that they constitute an orthonormal system. It is clear
20
1. Frames in Finite-dimensional Inner Product Spaces
that ||efc|| = 1 for all k. Now, given к
q) = — е27гг^ 27гг^ n П
Using the formula (1 — ж)(1 4- x 4- • • • 4- xn x) = 1 - xn with x = e2*4*"*, we get
11- (e2™—)n n i _ e2™—
□
The basis {e/JJE^ is called the discrete Fourier transform basis. Using this basis, every sequence f € O1 has a representation
f = Е(Л ek)ek =	E E f^e-2^-^ek.
k=l	k-1 £=1
Written out in coordinates, this means that
/(» =
n fc=i £=i
= -EE^We2’ri(J’'£)i^i> j =
k=l £=1
Applications often ask for tight frames because the cumbersome inversion of the frame operator is avoided in this case, see (1.9). It is interesting that overcomplete tight frames can be obtained in Cn by projecting the discrete Fourier transform basis in any Cm, m > n, onto Cn:
Proposition 1.4.2 Let m > n and define the vectors {fk}™-i in C1 by
\ e27r^(n-l)^il у
к = 1,2,..., m.
Then	is a tight overcomplete frame for C1 with frame bound equal
to one, and ЦД-Ц = for all k.
1.4 The discrete Fourier transform 21
proof. Let be the canonical basis for Cn, and let	be the
discrete Fourier transform basis for C™, i.e.,
e27ri(n-l)^il-
e27ri(m-l)^l- j
Identifying Cn with a subspace of , the orthogonal projection of e* onto Cn is Pek = fk', now the result follows from Exercise 1.9.	□
It is important to notice that all the vectors Д in Proposition 1.4.2 have the same norm. If needed, we can therefore normalize them while keeping a tight frame; we only have to adjust the frame bound accordingly.
Corollary 1.4.3 For any m > n, there exists a tight frame in C71 consisting of m normalized vectors.
Example 1.4.4 The discrete Fourier transform basis in C4 consists of the vectors
1
2
		(			
1	1	г 1 1	-1	1	—i
1	’ 2	—11’2	1	’ 2	-1
w					\»/
Via Proposition 1.4.2, the vectors
1 /1\ 1 A \ 1 f 1 \ 1 f 1 \
2 V/ ’ 2 V/ 2	2
constitute a tight frame in C2.
□
One advantage of an overcomplete frame compared to a basis is that the frame property might be kept if an element is removed. However, even for frames it can happen that the remaining set is no longer a frame, for example if the removed element is orthogonal to the rest of the frame elements. Unfortunately this can be the case no matter how large the number of frame elements is, i.e., no matter how redundant the frame is! If we have information on the lower frame bound and the norm of the frame elements we can provide a criterion for how many elements we can (at least) remove:
Proposition 1.4.5 Let {fk}^-! be a normalized frame for C71 with lower frame bound A > 1. Then, for any index set I C {1,... ,m} with |/| < A, the family {fk}k£i is a frame for C71 with lower bound A — |7|.
22
1. Frames in Finite-dimensional Inner Product Spaces
Proof. Given f e Cn,
Ek/,a>i2<I2iiaii2ii/ii2 = i^ ii/ii2-
k(=I	kel
Thus
Ек/,АИ2>(^-И)11/Н2-
k(£I
□
Theorem 1.2.1 shows that if {Д}^_1 is a tight normalized frame, then Proposition 1.4.5 applies if |Z| < Considering an arbitrary frame {A}£Li f°r Cn, the maximal number of elements one can hope to remove while keeping the frame property is m — n. If we want to be able to remove m - n arbitrary elements it is not enough to assume that {fk}™=1 is a normalized tight frame, as demonstrated by the frame in Example 1.3.5; in this example m — n = 2, but the three first vectors in (1.18) do not constitute a frame for C3. Concerning the stability against removal of vectors, the frames obtained in Proposition 1.4.2 behave well: m — n arbitrary elements can be removed:
Proposition 1.4.6 Consider the frame {fk}™^ for Cn defined in Proposition 1.4.2. Any subset containing at least n elements of this frame forms a frame for C71.
Proof. Consider an arbitrary subset {, A2,..., кп} С {1,2,..., m}. Placing the vectors {At}™=1 as rows in an n x n matrix and letting z := e~^ we obtain
/ -fk,- \ — fk2 —
\ -A„- /
.'(fcj-lXn-l) \ г(к2-1)(п-1)
—l)(n —1) j
this is a Vandermonde matrix with determinant
^2 П
Thus {At}?=i a basis for Cn by Theorem 1.3.1.	□
1.5 Pseudo-inverses and the singular value decomposition 23
1.5 Pseudo-inverses and the singular value decomposition
It is well known from linear algebra that not all matrices have an inverse. Keeping in mind how useful inverses are, it is natural to search for some types of “generalized inverses” in case no inverse exists, which capture at least some of the nice properties.
The right definition of a generalized inverse depends on the properties we are interested in, and we shall only define the so-called pseudo-inverse. Given an m x n matrix E, we consider it as a linear mapping of Cn into Crn. E is not necessarily injective, but by restricting E to the orthogonal complement of the kernel Me, we obtain an injective linear mapping
E:A/^ ->Cm.
E and E have the same range, K& = Ke', thus E considered as a mapping from to Ke has an inverse,
(Ё)-1 : Кд-> A/g.
We can extend (E)~l to an operator : C™ -> Cn by defining
E\y + 2^(ЁГ1уИуе-ЕЕ,ге HE.	(1.21)
With this definition,
EE]x = x, Mx&IZe-	(1-22)
The operator E^ is called the pseudo-inverse of E. From the definition we immediately have that
= лгЕ., tze, = v| = nE..	(i.23)
We note two characterizations of the pseudo-inverse:
Proposition 1.5.1 Let E be an m x n matrix. Then
(i) E^ is the unique n x m matrix for which EE^ is the orthogonal projection onto Ke and E^E is the orthogonal projection onto KEi-
(ii) E^ is the unique nxm matrix for which EE^ and E^E are self-adjoint and
EE]E = E, E^EE^ = Ef.
Proof. We first prove the equivalence between the conditions stated in (i) and (ii). If a matrix E^ satisfies (i), it immediately follows that (ii) is satisfied. On the other hand, if (ii) is satisfied, then
(EEf)2 = EE^EE^ = EE\
24
1. Frames in Finite-dimensional Inner Product Spaces
Since EE^ is self-adjoint it follows that EE^ is the orthogonal projection onto Eee^. Finally, the identity EE^E = E shows that Eee\ —Ee. The proof that E^E is the orthogonal projection onto is similar. Thus (i) is satisfied.
We now prove the equivalence between the properties in Proposition 1.5.1 and the definition (1.21) of the pseudo-inverse. First we note that with our definition of the pseudo-inverse, the conditions in (i) are satisfied; the main ingredients in the following argument are the relations (1.22) and (1.23). In fact, if у E Ee, then EE^y = y\ and if у E Ee = A/^t, then EE^y = 0. This proves that EE^ is the orthogonal projection onto KE. Also, if у E = Л/e, then E^Ey = 0; and if у E EEt, у = E^x for some x, then
E]Ey = E^EE^x = E^x - E\l -EE^x = E]x = y.
Here we used that I - EE^ is the orthogonal projection onto 1lE = AfEi. We have now proved that E^E is the orthogonal projection onto Ee\.
To conclude we only have to prove that if a matrix E^ satisfies (i) and (ii), then it fulfills the requirements in the definition of the pseudo-inverse, i.e., (1-21) is satisfied. First, we note that (ii) implies that
E* = (EjE)*E* = E^EE*;
this shows that
A/# = Ее* C E& .
Now, if у E Ее, then we can find x E AfE such that у = Ex\ thus
E'y = E^Ex = x = (E)~1Ex = (E)~1y.
Finally, if z E Ee = Me*, then by (i), EEh = 0; using (ii),
E^z = E^EE^z = 0.	□
The pseudo-inverse gives the solution to an important minimization problem:
Theorem 1.5.2 Let E be an m x n matrix. Given у E Ee, the equation Ex = у has a unique solution of minimal norm, namely x = E^y.
Proof. By (1.22), we know that x := E^y is a solution to the equation Ex = y. All solutions have the form x = E^y + z, where z E Ne- Since E^y e AfE, the norm of the general solution is given by
|H|2 = ||Etj/ + z||2 = ||Etj,||2 + |M|2>
which is minimal when z = 0.	□
1.5 Pseudo-inverses and the singular value decomposition 25
Historically (i) and (ii) were given as definitions of a “generalized inverse” by Moore, respectively Penrose. For this reason the pseudo-inverse is frequently called the Moore-Penrose inverse.
For computational purposes it is important to notice that the pseudoinverse can be found using the singular value decomposition of E. We begin with a lemma.
Lemma 1.5.3 Let E be an m x n matrix with rank r > 1. Then there exist constants ,..., ar > 0 and orthonormal bases {wfc}^=1 for Ее and for Ее* such that
Evk = &kuk, /c = l,...,r.	(1-24)
Proof. Observe that E*E is a self-adjoint n x n matrix; by Theorem A.2.1 this implies that there exists an orthonormal basis {vfc}J_1 for Cn consisting of eigenvectors for E*E. Let {А&}£_2 denote the corresponding eigenvalues. Note that for each к,
к = A/Jk-ll2 = <E*Evk,vk) = ЦЖ-Ц2 > 0.
The rank of E is given by
r = (ИшЕе = dimT^#*;
since Eg = Ne* , we have
Ee* = Ee*e = span{E* Evk}k=i = $Р&п{Хкик}к=1. (1.25)
Thus, the rank is equal to the number of non-zero eigenvalues, counted with multiplicity. We can assume that the eigenvectors {vfc}JL.x are ordered such that	corresponds to the non-zero eigenvalues. Then (1.25) shows
that	is an orthonormal basis for Ее*- Note that for к > г, we have
\\Evk\\2 = {E* Evk,vk) = 0, i.e,
Evk =0, к > r.	(1.26)
Defining
nk := —^=Evk, к = 1,..., r, vAfc
we therefore obtain that	spans Ее’, and it is an orthonormal basis
for Ее because for all k,I = 1,..., r we have
,Ul} = 7^7^Evk’
= ~rLr={E* Evk,vi)
26
1. Frames in Finite-dimensional Inner Product Spaces
Thus, the conditions in Lemma 1.5.3 are fulfilled with
°k = к = l,...,r.
□
Lemma 1.5.3 leads to the singular value decomposition of E:
Theorem 1.5.4 Every m x n matrix E with rank r > 1 has a decomposition
e=u(Dq J V*-	o-27)
where U is a unitary m x m matrix, V is a unitary n x n matrix, and (	0 ) 25 аП m X П Ы°ск matrix in which D is an r x r diagonal
matrix with positive entries &i,... ,ar in the diagonal.
Proof. We use the proof of Lemma 1.5.3. Let {u/JJL.j be the orthonormal basis for Cn considered there, ordered such that {ufc}^=1 is an orthonormal basis for 7Ze*- Let V be the n x n matrix having the vectors	as
columns. Extend the orthonormal basis {ufc}£=1 for TZe to an orthonormal basis	for and let U be the m x m matrix having these vectors
as columns. Finally, let D be the r x r diagonal matrix having <ti, ..., ar in the diagonal. Via (1.24) and (1.26),
(
EV
arur
0 • • 0 )
Multiplying with V* from the right gives the result.
The numbers ai,...,ar are called singular values for E; the proof of Lemma 1.5.3 shows that they are the square roots of the positive eigenvalues for E*E.
Corollary 1.5.5 With the notation in Theorem 1.5.4, the pseudo-inverse of E is given by
0
0
0
(1-28)
where	J is an n x m block matrix in which D 1 is the r x r
matrix having l/(Ti, ..., l/crr in the diagonal.
1.5 Pseudo-inverses and the singular value decomposition 27
Proof. We check that the matrix defined by (1.28) satisfies the requirements in Proposition 1.5.1(ii). First, via (1.27),
EE' = "(o S ) v'v ( Co"‘ S)1'"
which shows that EE^ is self-adjoint. The proof that E^E is self-adjoint is similar. Furthermore, using the derived expression for EE\
EE'E = u{‘o S)1"
= E.
Similarly, one can verify that E^EE* = EK
□
Let us return to the setting where	is a frame for C71 with pre-
frame operator T : Cm -> C1. The calculation of the frame coefficients amounts to finding TK
Theorem 1.5.6 Let {fk}™=1 be a frame for Cn, with pre-frame operator T and frame operator S. Then
ту = {</, s-1 a>}ZLi, V/ e e.	(1.29)
Proof. Let f E Cn. Expressed in terms of the pre-frame operator T, the equation f = SfcLi ckfk means that T^k}™-! = f. The result now follows by combining Theorem 1.1.5 and Theorem 1.5.2.	□
One interpretation of Theorem 1.5.6 is that when {fk}™-i is a frame for Cn, the matrix for is obtained by placing the complex conjugate of the vectors in the canonical dual frame {S-1 fk}^-i as rows in an m x n matrix:
=
-s-71- \
-s-72-
\ -S-'f™- /
In operator terms, (1.29) means that
T^ = T* (tt*)-1
a formula that is known to hold generally for the pseudo-inverse of an arbitrary surjective operator T.
The singular value decomposition gives a natural way to obtain coefficients	such that f = Y?k=i ckfk- Let {fk}™-i be an overcomplete
28
1. Frames in Finite-dimensional Inner Product Spaces
frame for Cn and let f E O1. Since T is surjective, its rank equals n, and the singular value decomposition of T is
T = U( D 0 ) V\
Note that since T is an n x m matrix, ( D 0 ) is now an n x m matrix; U is an n x n matrix, and V is an m x m matrix. Given any (m — n) x n matrix F, we have
TV^DF^U*f = U(D 0 ) V*V ( DFl U*f = UIU*f = /•
This means that we can use the coefficients
{MZU = v ( df ) u*f
for the reconstruction of f. As noted already in Theorem 1.1.5 the choice F = 0 is optimal in the sense that the £2-norm of the coefficients is minimized, but for other purposes other choices might be preferable. The matrix
is frequently called a generalized inverse of T.
1.6 Finite-dimensional function spaces
The rest of the book will deal with frames in infinite-dimensional vector spaces, with concrete constructions in function spaces like L2(—7г,7г) and L2(K); their exact definition will be given in Chapter 2, and for the moment we consider L2(Z),/ С К simply as the set of functions for which
|/(x)|2rfx < oo.
It is important to notice that in every real-life application where these spaces appear, one will at some point have to confine to finite-dimensional subspaces. For this reason we conclude this chapter with a short description of frames in finite-dimensional function spaces.
Given a, b E К with a < b, let C[a, b] denote the set of continuous functions / : [a, 5] -> C. We equip C[a, 5] with the supremums-norm, ll/lloo = sup |/(z)| .
The Weierstrass’ Approximation Theorem says that every f E C[a,5] can be approximated arbitrarily well by a polynomial:
1.6 Finite-dimensional function spaces
29
Theorem 1.6.1 Let f € C[a, b]. Given e > 0, there exists a polynomial P(x) = ckxk such that
\\f-P\\oo<e.
It is essential for the conclusion that [a, b] is a finite and closed interval (Exercise 1.12). Also, we note that the order of the approximating polynomial depends as well on the chosen e as the given function f and the interval [a, b].
The polynomials {1, x, x2,... } =	are linearly independent and
do not span a finite-dimensional subspace of C[a, b], But for a given n E N, the vector space
V := span{l, x,... , zn}
is a finite-dimensional subspace of C[a, b] with the polynomials {Tfc}^_0 as basis.
If we equip V with the || • Цоо-norm, we do not have the benefit of a norm arising from an inner product. But all norms on a finite-dimensional vector space are equivalent (see page 13), and V can also be equipped with the norm
/ гь	\1/2
11/11 = / l/«<fc
\ a	/
arising from the inner product
(f,9}= [ f(x)g(x)dx.	(1.30)
J a
Via the Gram-Schmidt orthogonalization procedure one can construct an orthonormal basis for (V, || • ||) (Exercise 1.14).
In classical Fourier analysis one expands functions in L2(0,1) in terms of the complex exponential functions {e2irlkx}kez- In Chapter 7 we will obtain more general results with {e2nikx}ke% replaced by {elXkX}kez for some real sequence {Xk}kez satisfying certain density conditions. Let us for the moment consider a finite collection of exponential functions	,
where {Afc}J?=1 is a sequence of real numbers. Unless {Afc}£=1 contains repetitions, such a family of exponentials is always linearly independent:
Lemma 1.6.2 Let {А&}£=1 be a sequence of real numbers, and assume that Xk Xj for к 7^ j. Let I C R be an arbitrary non-empty interval, and consider the complex exponentials {elAfcX}^_1 as functions on I. Then the functions [егХкХare linearly independent.
Proof. It is enough to prove that the functions {elXkX}kez are linearly independent as functions on any bounded interval ]a, &[, where a,6 G R,
30
1. Frames in Finite-dimensional Inner Product Spaces
a < b. Assume that for some coefficients {cfcjJUn
^2сьегХкХ = 0, Vz e]a,b[. k=i
When x runs through the interval	the variable x 4- runs
through ]a, 6[; it follows that
£CfceUfc(x+^)=0, уж€]^,^[.
Writing dk '= с^егХк а?Ь this leads to
k=l
By differentiating this equation j times, j = 0,1, • • •, we obtain that
£4(iAfcyeiA^ = o,	j = o,i,---.
fc=l
Putting x = 0 and writing the corresponding equations for j = 0,..., n — 1 as a matrix equation gives
( 1 1
/ dl \ d^
\ dn j
The system matrix is a Vandermonde matrix with determinant
n
Д= П (Ak-AJ/0;
therefore d\ = d2 = • • • = dn = 0, which implies that c\ = • • • = cn = 0.
Thus {e1Afc:c}^_1 are linearly independent.	□
In words, Lemma 1.6.2 means that complex exponentials do not give natural examples of frames in finite-dimensional spaces: if Xk ф X3 for к ф j, then the complex exponentials {elAfcI}^_1 form a basis for their span in L2(I) for any interval I of finite length, and not an overcomplete system. We can not obtain overcompleteness by adding extra exponentials (except by repeating some of the Л-values) - this will just enlarge the space. In Exercise 1.15 the similar problem for sines and cosines is considered.
1.6 Finite-dimensional function spaces 31
As an important special case we now consider the case where = 2тгк. A function f which is a finite linear combination of the type
/(ж) = cke21rlkx for some ck e C, NltN2 eZ,N2> M (1.31) k=Ni
is called a trigonometric polynomial. Trigonometric polynomials correspond to partial sums in Fourier series, a topic to which we return in Section 3.7. A trigonometric polynomial f can also be written as a linear combination of functions sin(27rA;j:), cos(2tfA;j:), in general with complex coefficients. It will be useful later to note that if f is real-valued and the coefficients ck in (1.31) are real, then f is a linear combination of functions cos(2ttA;z) alone:
Lemma 1.6.3 Assume that the trigonometric polynomial f in (1.31) is real-valued and that the coefficients Ck G IR. Then
n2
f(x) = Ck cos(2ttA;z).	(1.32)
We leave the short proof to the reader. Note that we need the assumption that Ck G 1R: for example, the function
№) = leHx _ l_e-™ = sin(;r)j
is real-valued, but does not have the form (1.31).
Again for later use we mention that a positive-valued trigonometric polynomial has a square root (in the sense of (1.35) below), which is again a trigonometric polynomial:
Lemma 1.6.4 Let f be a positive-valued trigonometric polynomial of the form
N
f(x) = cos(2ttH), Ck E IR.	(1.33)
k=0
Then there exists a trigonometric polynomial
N
g(x) = У2 ^ке27ггкх with dk E JR,	(1-34)
k=o
such that
= /(x), Vx G R.	(1.35)
A constructive proof can be found in [106]. Note that by definition, the function g in (1.34) is complex-valued, unless f is constant; that the complex terms in g do not cancel follows from Exercise 1.15. Actually, despite
32
1. Frames in Finite-dimensional Inner Product Spaces
the fact that f is assumed to be positive, there might not exist a positive trigonometric polynomial g satisfying (1.35).
The complex exponentials do not belong to L2(K), but by multiplying them with a function g € L2(K) we obtain a class of functions in L2(K). In Chapters 8-10 we will work with systems of functions in L2(K) of the form {ЕтьТпад}т,п& ~ {e2nmbxg(x-na)}m,n^
here g is a given function in L2(K), and the parameters a, b are positive real numbers. Such a family of functions is called a Gabor system. It was proved by Linnell [214] in 1997 that if g 0, then an arbitrary finite subfamily {e2™mbxg(z-na)}(m>n)G5-, У c Z2, is linearly independent. At the moment it is not known what happens if we replace the numbers {(na, m6)}m>nGz by arbitrary distinct points in K2. To be more precise, Heil, Ramanathan and Topiwala [170] formulated the following conjecture in 1995:
Conjecture: Given any finite collection of distinct points {(//£,Xk)}kQ^ in tf2 and a function g ± 0, the Gabor system {e27riAfcXg(z - Hk)}ke^ is linearly independent.
Considerable effort has been invested in the conjecture, but it is still open. We return to this conjecture in its right context on page 229. Also, in Section 10.3 we will construct frames in Cn having the Gabor-structure.
Wavelets is another important class of functions in L2(K); we consider them in detail in Chapters 11-14. A wavelet system consists of functions of the form
= 2J/2^(2J\r — /с), к e Z,
where € L2(K) is a given function. Linearly dependent wavelet systems exist. For example, by letting := X[o,i[> one has
V’o.o = \/2^1,C> +
If a finite wavelet system, {ipj,k}p|,\k\<N for some N e N, happens to be linearly independent, one could ask for the minimal number of independent sets it can be split into. One could expect to grow with N; however, in case has compact support and |^| > 0 on some interval of positive length, it is proved in [83] that one can find a number m G N such that {ф^к}p|,|fc|<N can be split into m linearly independent sets, regardless of how large N is. It is not known whether the result holds if ip is not assumed to have compact support.
1.7 Exercises
1.1	Show that every frame {fk^-i for a finite-dimensional vector space V contains a subset which is a basis for V.
1.7 Exercises 33
1.2	Can a frame in a finite-dimensional space contain infinitely many elements?
1.3	Let {fk}kei be a frame for a finite-dimensional vector space V and assume that ||Д|| is bounded below. Prove that I is finite, (w.l.o.g. you may assume that V = Жп and that ||Д|| = 1, V/c; explain why if you want to use this fact!)
1.4	Construct a frame {fk}™=i for C2 for which there exists / G C2 such that the coefficients {dk}™=1 in Theorem 1.1.7 are not unique.
1.5	Let {61,62} be the canonical orthonormal basis for C2 and consider the frame {fk}3k=i = {^i, e2, ex 4- e2}.
(i)	Find the coefficients with minimal £2-norm among all sequences {MLifor which ei = ELi cfcA-
(ii)	Find the coefficients {cj^JiUi an<^ {cfe2,}'fe=i which minimize the £r-norm in the representation of ei and e2, respectively.
(iii)	Clearly, a + e2 = Ski (ck'> + cfc2’)A; but is {c[1} + c^2)}Li minimizing the ^-norm among all sequences representing ei 4- e2?
1.6	Assume that {fk}™=i is a frame for CL Prove that the 2m vectors consisting of the real parts, respectively the imaginary parts, of the frame vectors constitute a frame for .
1.7	Show that a frame for is also a frame for Cn.
1.8	Prove Corollary 1.3.6.
1.9	Let {fkYkLi be a frame for V with bounds A, В and let P denote the orthogonal projection of V onto a subspace W. Prove that {F/fc}^Li is a frame for W with frame bounds A, B.
1.10	Let {Л)ь=1 be a normalized tight frame. Prove that the frame bound A is at least 1, and that A = 1 if and only if {fk}™=i is an orthonormal basis.
1.11	Let {fk}™-i be a frame for an n-dimensional vector space V, and let В denote the optimal upper bound. Prove that
m
В<^Ы<пВ.
k~l
34
1. Frames in Finite-dimensional Inner Product Spaces
1.12	Prove that Theorem 1.6.1 fails if the closed and bounded interval [a, b] is replaced by an open interval or an unbounded interval.
1.13	Prove that for any n 6 N, the polynomials {1,	, xn} are
linearly independent in C(0,1).
1.14	Consider the polynomials {1, z, x2} as functions on the interval [0,1], and let V = span{l, x, x2}. Equip V with the inner product (1.30) and find an orthonormal basis for V.
1.15	Let	be a sequence of real numbers.
(i)	Prove that {cos A^rr}^! are linearly independent in C(—1,1) if and only if |Afc|	|Xj| for к 7^ j.
(ii)	Prove that {sin AjczJJLj are linearly independent in C(—1,1) if and only if all A^ are non-zero and |Afc| Ф |Xj| for к 7^ j.
(iii)	Under which conditions on sequences {Afc}^=1, {м/JkLi are the functions
{созА£ж}£=1 U {sin/^x}^!
linearly independent in C(—1,1)?
(iv)	Replace the interval ] — 1,1[ by an arbitrary non-empty interval and generalize (i),(ii) and (iii).
1.16	Consider the positive trigonometric polynomial f(x) = 1 4"Cos(t). Find by direct calculation all trigonometric polynomials
g(x) = do 4- die™, do,di 6 R, for which \g(x) |2 = f(x).
2
Infinite-dimensional Vector Spaces and Sequences
After the introduction to frames in finite-dimensional vector spaces in Chapter 1, the rest of the book will deal with expansions in infinitedimensional vector spaces. Here great care is needed: we need to replace finite sequences {fk}k==i by infinite sequences {fk}fc=i, and suddenly the question of convergence properties becomes a central issue. The vector space itself might also cause problems, e.g., in the sense that Cauchy sequences might not be convergent. We expect the reader to have a basic knowledge about these problems and the way to circumvent them, but for completeness we repeat the central themes in Sections 2.1-2.2. In Sections 2.3-2.5 we discuss the Hilbert space L2(K) consisting of the square integrable functions on К and three classes of operators hereon, as well as the Fourier transform. The material in those sections is not needed for the study of frames and bases on abstract Hilbert spaces in Chapters 3-6, but it forms the basis for all the constructions in Chapters 7-14.
2.1 Sequences
A central theme in this book is to find conditions on a sequence {fk} in a vector space X such that every f G X has a representation as a superposition of the vectors Д. In most spaces appearing in functional analysis, this can not be done with a finite sequence {Л}. We are therefore forced to work with infinite sequences, say, {fk}^-^ and the representation of f in terms of {fk}kLi will be via an infinite series. For this reason the
36
2. Infinite-dimensional Vector Spaces and Sequences
starting point must be a discussion of convergence of infinite series. We collect the basic definitions here together with some conventions.
Throughout the section we let X be a normed vector space, with norm denoted by ||-||. We say that a sequence }^_1 in X
(i) converges to x e X if
| |rr — Zfc|| -> 0 for к -> oo;
(ii) is a Cauchy sequence if for each б > 0 there exists N 6 N such that
|\xk — xi11 <6 whenever k,l > N.
A convergent sequence is automatically a Cauchy sequence, but the opposite is not true in general. There are, however, normed vector spaces in which a sequence is convergent if and only if it is a Cauchy sequence; a space X with this property is called a Banach space. All spaces considered in this book are Banach spaces.
Imitating the finite-dimensional setting, we want to study sequences {fk}kLi in X with the property that each / G X has a representation f = Skii ckfk for some coefficients Ck G C. In order to do so, we have to explain exactly what we mean by convergence of an infinite series, and there are, in fact, at least three different options. First, the notation {fk}kLi indicates that we have chosen some ordering of the vectors Д,
/1, /2, /3, ♦ • •, A, fk+i, • • • •
We say that an infinite series 52 ^=1 Ckfk is convergent with sum f G X if
n
f - Ск^к
к=1
—> 0 as n —> 00.
When this condition is satisfied we write
00
/ = Ecfc/fc.	(2.1)
k=l
Thus, the definition of a convergent infinite series corresponds exactly to our definition of a convergent sequence with xn = 52^—1 ckfk-
Above we insisted on a fixed ordering of the sequence {A}£Li ♦ is уегУ important to notice that convergence properties of 52kLi cfcA not only depends on the sequence {A}j£i and the coefficients {ck}^=11 but also on the ordering. Even if {A}£i is a sequence in the simplest possible Banach space, namely R, it can happen that 52^=1 A is convergent, but that 52/Xi A(fc) is divergent for a certain permutation a of the natural numbers. This observation leads to a second definition of convergence. If 52 ^1 A(fc) is convergent for all permutations cr, we say that 52aXi A is unconditionally convergent. In that case, the limit is the same regardless of the order of summation.
2.1 Sequences 37
As soon as we have defined frames and Riesz bases it will become clear that they automatically lead to unconditionally convergent expansions. For this reason we never need to prove by hand that a given series converges unconditionally. We refer to [210] and [260] for a more detailed analysis of the different types of convergence and the proof of the following lemma.
Lemma 2.1.1 Let {fk}kLi be a sequence in a Banach space X, and let ‘ f E X. Then the following are equivalent:
(ii) fk converges unconditionally to f G X.
(ii) For every € > 0 there exists a finite set F such that
< e
kei
for all finite sets I C N containing F.
Finally, an infinite series A is said to be absolutely convergent if
oo
EllAII<oo.
Absolute convergence of fk implies that the series converges unconditionally (Exercise 2.3), but the opposite does not hold in infinitedimensional spaces. In finite-dimensional spaces the two types of convergence are identical.
A subset Z С X (countable or not) is said to be dense in X if for each f G X and each б > 0 there exists g G Z such that
II/-fill <e-
In words, this means that elements in X can be approximated arbitrarily well by elements in Z.
For a given sequence {A}£Li m X we let sPan{A}^=i denote the vector space consisting of all finite linear combinations of vectors Д. The definition of convergence shows that if each f G X has a representation of the type (2.1), then each f G X can be approximated arbitrarily well in norm by an element in spanf/^}^, i.e.,
span{/fc}^=i=X	(2.2)
A sequence {A}£Li having the property (2.2) is said to be complete or total. We note that there exist normed spaces where no sequence {A}£Li is complete. A normed vector space in which a countable and dense family exists is said to be separable.
When we speak about a finite sequence, we mean a sequence where at most finitely many entries are non-zero.
38	2. Infinite-dimensional Vector Spaces and Sequences
2.2 Banach spaces and Hilbert spaces
All normed vector spaces considered in this book are Banach spaces, and very often convergence of a sequence will be verified by checking that it is a Cauchy sequence.
An important class of Banach spaces is the Lp-spaces, 1 < p < oo. L°°(R) is the space of essentially bounded measurable functions f : К -» C, equipped with the supremums-norm. For 1 < p < oo, LP(K) is the space of functions f for which \f\p is integrable with respect to the Lebesgue measure:
LP(K) := | f : К -> C | / is measurable and /* |/(n;)|р</гг < oo| .
I	J —oo	J
The norm on LP(1R) is
/ roo	\ i/p
11/11= / \f^\pdx) .
\J — oo	/
To be more precise, LP(K) consists of equivalence classes of functions which are equal almost everywhere, and for which a representative (and hence all) for the equivalence class satisfies the integrability condition. However, we adopt the standard terminology and speak about functions in Lp(K).
A vector space X with an inner product (•, •) can be equipped with the norm
||z|| := У(z,x), x € X,
and Cauchy-Schwarz ’ inequality states that
IWI<IHI ||< \fx,yeX.
We will always choose the inner product linear in the first entry. A vector space with inner product, which is a Banach space with respect to the induced norm, is called a Hilbert space. We reserve the letter H for these spaces. The standard examples are the spaces L2(K) and ^2(N) discussed in the next section.
2.3 L2(B) and £2(N)
L2(K) is defined as the space of complex-valued functions, defined on K, which are square integrable with respect to Lebesgue measure:
L2(JR) := If : К -> C | / is measurable and /* |/(z)|2dz < oo
I	J — oo
2.3 L2(R) and £2(N)
39
£2(R) is a Hilbert space with respect to the inner product
Zoo ___________
f(x)g(x)dx, f,g&L2(K).
-oo
The spaces L2(Q), where Q is an open subset of R are defined similarly. According to the general definition, a sequence of functions	in
L2(Q) converges to g e L2(Q) if
/ r	\l/2
\\g-9k\\=\ IpW - gk(x)\2dx )	-4 0 as к -4 oo.
\Jti	/
Convergence in L2 is very different from pointwise convergence. As a positive result we have Riesz" Subsequence Theorem:
Theorem 2.3.1 Let Q C R be an open set, and let {gk} be a sequence in L2(Q) which converges to g G L2(Q). Then {gk} has a subsequence {gnk}^i such that
g(x) = lim gnt(x)
k—>oo
for almost every x G Q.
The result holds no matter how we choose the representatives for the equivalence classes. This is typical for this book, where we rarely deal with a specific representative for a given class. There are, however, a few important exceptions. When we speak about a continuous function, it is clear that we have chosen a specific representative, and the same is the case when we discuss Lebesgue points. By definition, a point у G R is a Lebesgue point for a function f if
1 ГУ+h
Jim- J г l№) - f(x)\dx = 0.
If f is continuous in y, then у is a Lebesgue point (Exercise 2.1). More generally, one can prove that if f G L1 (R), then almost every у G R is a Lebesgue point.
It is clear from the definition that different representatives for the same equivalence class will have different Lebesgue points: for example, every у G R is a Lebesgue point for the function f = 0; changing the definition of f in a single point у will not change the equivalence class, but у will no longer be a Lebesgue point. See Exercise 2.1 for some related observations.
In L2(R), Cauchy-Schwarz’ inequality states that for all f,g G L2(R),
Zoo	/ roo	\	1/2 / roo	\	1/2
(/	1рО)|2<7ж)	. z
-OO	\«/— OO	/ v-oo	/
40	2. Infinite-dimensional Vector Spaces and Sequences
The discrete analogue of L2(K) is ^2(Z), the space of square summable scalar sequences with a countable index set I:
:=|pfe}fc6/CC| £Ы2 < oo
I	I
^2(Z) is a Hilbert space with respect to the inner product
<{sk}, {yk}) =^xky^;
kei
in this case Cauchy-Schwarz’ inequality gives that
2
< 52ы2^2ы2, {Vkjkei 6 ?(I).
kei	kei kei
2.4 The Fourier transform
For f G the Fourier transform f is defined by
/(7) == Г f(x)e~2^dx, 7 e R.
J —oo
Frequently we will also denote the Fourier transform of f by Ff.
If (L1 П L2)(K) is equipped with the L2(K)-norm, the Fourier transform is an isometry from (L1 HL2)(K) into L2(K). If f G L2(K) and {fkYkLi is a sequence of functions in (L1 П L2)(K) which converges to f in L2-sense, then the sequence {fk}^-i is also convergent in L2(K), with a limit which is independent of the choice of {fkYkLi - Defining
/ := lim fk к—too
we can extend the Fourier transform to a unitary mapping of L2 (R) onto L2(K). We will use the same notation to denote this extension. In particular we have Plancherel’s equation
<f.9) = {f,9\ Vf,9t b2TO, and H/Ц = ||/||.	(2.3)
If f G L1(K), then f is continuous. If the function f as well as f belong to Z,1 (1R), the inversion formula describes how to come back to f from the function values /(7):
Theorem 2.4.1 Assume that f,f€ Then
f(x) = [°°	a.e. x € K.	(2.4)
J —oo
The pointwise formula (2.4) holds at least for all Lebesgue points for f, cf. [10].
2.5 Operators on L2(R)	41
2.5 Operators on L2(R)
In this Section we consider three classes of operators on L2(R) which will play a key role in our analysis of Gabor frames and wavelets. Their definitions are as follows:
Translation by a EK, Ta : L2(R) -> L2(R), (Taf)(x) = f(x — a); (2.5) Modulation by b E K, Eb : L2(R) -» L2(R), (Eb/)(z) = е27гг^/М; (2-6)
Dilation by a / 0, Da : £2(R) -4 L2(R), (Dof)(x) = -2=/(-). (2.7) vH a
A comment about notation: we will usually skip the brackets and simply write Taf(x), and similarly for the other operators. Frequently we will also let Eb denote the function x i-> е2пгЬх. We collect some of the most important properties for the operators in (2.5)-(2.7):
Lemma 2.5.1 The translation operators satisfy the following:
(i) Ta is unitary for all a G R.
(ii) For each f G L2(R); у н-> Tyf is continuous from R to L2(R).
Similar statements hold for Eb,b e R and Da, а ф 0.
Proof. Let us prove that the operators Ta are unitary. Since
(Te/,5) = (/,T_a5), V/,5eL2(K),
we see that T* = T_a. On the other hand, Ta is clearly an invertible operator with T”1 = T_a, so we conclude that T”1 = T*.
To prove the continuity of the mapping у i-> Tyf we first assume that f is continuous and has compact support, say, contained in the bounded interval [c, d\. For notational convenience we prove the continuity in yQ = 0. First, for у G] —	|[ the function
ф(т) = Tyf(x) - Tyof(x) = f(x -y)- f(x)
has support in the interval [— 14- c, d+ Since f is uniformly continuous, we can for any given € > 0 find 5 > 0 such that
\f(x — y) — f(x)| < б for all x G R whenever |j/| < 5;
with this choice of 5 we thus obtain that
/ \ 1/2
\\Tyf-TyQf\\ = (f2 \f(x~y)-f(x)\2dx\
< б л/d - c + 1.
This proves the continuity in the considered special case. The case of an arbitrary function f G L2(R) follows by an approximation argument, using
42	2. Infinite-dimensional Vector Spaces and Sequences
that the continuous functions with compact support are dense in L2{1R) (Exercise 2.4). The proofs of the statements for Еъ and Da are left to the reader (Exercise 2.5).	□
Chapters 8-14 will deal with Gabor systems and wavelet systems in L2(JR); both classes consist of functions in L2(IR) which are defined by compositions of some of the operators Ta, Еъ and Da. For this reason the following commutator relations are important:
TaEbf(x)	=	e~^zbaEbTJ(x) =e^b(x-a'>f(x-ah	(2.8)
TbDaf(x)	=	DaTb/af(x) = -}=f	(2.9)
л/|а| a a	f
DaE„f(x)	=	-1=е2^/а/(-) = ЕьСв/(х).	(2.10)
Vl«l a
In wavelet analysis the dilation operator D1^2 plays a special role, and we simply write
Df(x) := 21/2/(2x).
With this notation, the commutator relation (2.9) in particular implies that
TkDj = D3T2Jk and DjTk = T2-]kDj, j, keZ. (2.11)
We will often use the Fourier transformation in connection with Gabor systems and wavelet systems. In this context we need the commutator relations
ETa = E_aE, EEa-TaE, EDa = Dl/aT,	(2.12)
2.6 Exercises
2.1	Here we ask the reader to prove some results concerning Lebesgue points.
(i)	Assume that f : К -» C is continuous. Prove that every у G JR is a Lebesgue point.
(ii)	Prove that x = 0 is not a Lebesgue point for the function X[o,i] •
(iii)	Let / = xq. Prove that every у £ Q is a Lebesgue point, and that the rational numbers are not Lebesgue points.
2.2	Find a sequence	of real numbers for which ak is
convergent, but not unconditionally convergent.
2.6 Exercises 43
2.3	Let {fk¥k=i be a sequence in a Banach space. Prove that absolute convergence of i A implies unconditional convergence.
2.4	Complete the proof of Lemma 2.5.1 by showing the continuity of У Tvf for f e L2(R).
2.5	Prove the statements about Еь and Da in Lemma 2.5.1.
2.6	Prove the commutator relations (2.12).
3
Bases
Bases play a prominent role in the analysis of vector spaces, as well in the finite-dimensional as in the infinite-dimensional case. The idea is the same in both cases, namely to consider a family of elements such that all vectors in the considered space can be expressed in a unique way as a linear combination of these elements. In the infinite-dimensional case the situation is complicated: we are forced to work with infinite series, and different -concepts of a basis are possible, depending on how we want the series to converge. For example, are we asking for the series to converge with respect to a fixed order of the elements (conditional convergence) or do we want it to converge regardless of how the elements are ordered (unconditional convergence)? We define the relevant types of bases in general Banach spaces in Section 3.1, but besides this we mainly consider Hilbert spaces. In Section 3.4 we discuss the most important properties of orthonormal bases in Hilbert spaces; we expect the reader to have some basic knowledge about this subject. A slight (but useful) modification leads to the definition of Riesz bases, which are treated in detail in Section 3.6. Orthonormal bases and Riesz bases satisfy the so-called Bessel inequality, which is the key to the observation that they deliver unconditionally convergent expansions and can be ordered in an arbitrary way. Sequences satisfying the Bessel inequality are therefore discussed already in Section 3.2.
Concrete examples of bases in function spaces are given in Sections 3.7 and 3.8, where the basic theory for Fourier series is revisited (again this subject is expected to be known) and Gabor bases as well as wavelet bases for L2(]R) are introduced. These sections form the background for Chapters 7-14.
46
3. Bases
In the entire chapter, X denotes a Banach space, and H is a Hilbert space with the inner product (♦, •) linear in the first entry. We will assume that the spaces are separable and infinite-dimensional, and we leave the modifications in the finite-dimensional case to the reader.
3.1 Bases in Banach spaces
The most fundamental concept of a basis was introduced by Schauder [253] in 1927. It takes place in a Banach space X, and captures the basic idea of having a family of vectors with the property that each f 6 X has a unique expansion in terms of the given vectors. All bases considered in this book are Schauder bases.
Before giving the formal definition, we emphasize once more that a sequence	in X is an ordered set, i.e.,
{ek}kLi = {ei,e2, • • }•
Definition 3.1.1 Let X be a Banach space. A sequence of vectors belonging to X is a (Schauder) basis for X if, for each f EX, there exist unique scalar coefficients {c/c(/)}^_1 such that
oo
J = £ct(/h.	(3.i)
Sometimes we refer to (3.1) as the expansion of f in the basis
Equation (3.1) merely means that the series f = 52£Li Ck(f)ek converges with respect to the chosen order of the elements. If the series (3.1) converges unconditionally for each f 6 X, we say that {е/с}^ is an unconditional basis. One can prove that {e^}£ТХ is an unconditional basis if and only if {е<т(к)}£1 is a basis for every permutation a of N, cf. [260]. In other words, if	is a basis which is not unconditional, there exists a permutation
a for which	is not a basis. It is known that every Banach space
which has a basis also has a conditional basis, cf. [233].
Besides the existence of an expansion of each f E X, Definition 3.1.1 asks for uniqueness. This is usually obtained by requiring	to be
independent in an appropriate sense. In infinite-dimensional Banach spaces, different concepts of independence exist:
Definition 3.1.2 Let {fkYj^i be a sequence in X. We say that
(i) {fk}kLi is linearly independent if every finite subset of {fk}kLi is linearly independent;
(H) {/fc}£i is ш-independent if whenever the series 52S=i is convergent and equal to zero for some scalar coefficients	then
necessarily Ck = 0 for all к G N.
3.1 Bases in Banach spaces
47
%iii) {/feltei is minimal if f, $ span{fk}k^, \/j 6 N.
The relationship between the definitions is as follows:
Lemma 3.1.3 Let {fk}kLi be a sequence in X. Then the following holds:
(i) Ц {fk}kLi is minimal, then {A}^ is ш-independent.
(ii) V {fk}kLi is ш-independent, then {fk}^-i is linearly independent.
The opposite implications in (i) and (ii) are not valid.
proof. For the proof of (i), assume that {A}^_1 is not cj-independent. Choose scalar coefficients {q}^_1 with Cj 0 for some j, such that
Ckfk = 0; then f, =	implying that f, G span{/k}k#J.
That is, {fk}kLi is not minimal. The statement (ii) is obvious, and the fact that the opposite implications are not valid is demonstrated by examples in Exercise 3.4.	□
A Banach space having a basis is necessarily separable. Most of the known separable Banach spaces have a basis; the first example of a separable Banach space not having a basis was constructed by Enflo [122] in 1972.
It is clear that a basis for X is complete and consists of non-zero vectors. Adding an extra condition leads to a characterization of bases:
Theorem 3.1.4 A complete family of non-zero vectors	in X is
a basis for X if and only if there exists a constant К such that for all m,n E N with m < n,
m
k=l
n
/c=l
(3-2)
< К
for all scalar-valued sequences {q}^
Proof. Suppose that {е/с}^=1 is a basis. Then each f € X has a unique expansion f = Y)kLi скек, and
lll/lll := sup n
< oo.
k=l
Note that if |||/||| = 0, then cfcefcll = 0 f°r u C N; it follows that Ck = 0 for all к 6 N, and f = 0. One can check (Exercise 3.1) that HI • HI satisfies the other conditions for a norm on X, and that X is a Banach space with respect to this norm. By definition of ||| • |||, we have 11/II < 111/|11, Vf 6 X, meaning that the identity operator is a continuous and injective mapping of (X, ||| • |||) onto (X, || • ||). By Theorem A.5.2 it follows that this operator has a continuous inverse, i.e., that there exists a constant К > 0 such that |||/||| < К ||/|| for all f 6 X. In particular, fixing
48
3. Bases
an arbitrary n G N and considering f =	we obtain (3.2). For
the other implication, assume that a complete family	of non-zero
vectors satisfies (3.2). Let A denote the vector space consisting of all f EX which can be expanded as f = Ckek for some coefficients {cfc}^. First we prove that A = X- since	is assumed to be complete we
know that A is dense in X, so it is enough to prove that A is closed. Let f G X, and choose a sequence	C A such that fj -» f as j -» oo.
Write fg = JZfcLi for appropriate coefficients	By (3.2), for
each i G N and all n > m > i, we have for all j, I G N that
l<₽-4°l INI < К
(3.3)
Given e > 0, choose N G N such that
By letting n -> oo, it follows from the above estimate that |4J) - 4°l IM < « for all г £ N, j,l> N, and, via the intermediate step (3.3),
m
Е(4Я-<Н
k=l
< € for all m G N, j, I > N.
(3-4)
(3.5)
For each i G N, the sequence	is convergent by (3.4), say, c-^ -» Ci
as I -» oo. By letting I -> oo in (3.4) and (3.5), we obtain that
|c^ - Ci\ I\ег 11 < 6 for all i G N, j>N,	(3.6)
and
m
E (4я
Ck^ ek
< e for all m G N, j > N.
(3.7)
3.1 Bases in Banach spaces 49
Kow, for given m G N and all j G N,
ТП
f - ^Cfcek
/c=l
< II/-All
It follows from here that 52/Xi 0^ converges to f. In fact, for a given б > 0 we can choose N G N so that (3.7) holds. By fixing a sufficiently large value for j > N we obtain that Ц/ — fj\\ < e; after that, we can obtain that ||/j - 52/Xi с1^е^|| - 6 by choosing m G N sufficiently large. Thus Ц/ — 521~i ckek11 < 36 for m sufficiently large. We conclude that / = 52S=i	i.e., / E Л as desired. To prove that {e^}£ТХ is a basis
we only need to show that if 52X1 ckek = 0, then q = 0 for all к G N. This again follows from (3.2): if 52iJ=i скек = 0, then for each i G N and all n > i,
led lledl < К
n
^скек
к=1
from here we obtain the result by letting n -> oo.
Theorem 3.1.4 is often formulated using the basis constant, which for an arbitrary sequence {е/г}^_1 is defined by
{771	П	'j
|| скек\| : m < n, 1|5>е*|| = i [.	(3.8)
k=l	k=l	J
If {ek}^ is a basis, this is clearly the smallest constant that can be used in (3.2). On the other hand, if the basis constant is infinite, then {e/JjgLj is not a basis. For a finite sequence	the basis constant is defined
as above, with the addition that we consider n < N.
The basis constant К tells whether the sequence {ek}£TX can be a basis with respect to the chosen order of the elements. We note that a similar characterization of unconditional bases exists, cf. [260]: a complete sequence {efc}£i consisting of non-zero elements is an unconditional basis if and only if its unconditional basis constant
SUP {ll	' Il52c*ejt|| = 1 euid = ±l,Vfc}
is finite.
Given a basis	it is clear that the coefficients {с/е(/)}^=1 in
(3.1) depend linearly on /. The mappings / -> ck(f) are called coefficient functionals. As a consequence of Theorem 3.1.4 they are continuous:
50
3. Bases
Corollary 3.1.5 The coefficient functionals	associated to a basis
{efc}£i for X are continuous, and can thus be considered as elements in the dual X*. If there exists a constant C > 0 such that ||efc|| > C for all к 6 N, then the norms of {cfcjjg-j. are uniformly bounded.
Proof. We use Theorem 3.1.4 and the notation introduced there. Given f G X, write f — Ck(f)ek- Then, for any j G N and all n > j,
МЛ I I IM <K
n
k=l
Letting n -> oo we obtain that
□
See Exercise 3.2 for the case where {e^}is not norm-bounded below.
A sequence {fk^-i in X and a sequence	in X* are said to be
biorthogonal if
— J1 if =
9k\fj) — °k,j •— \ n .f . , .
10 if
(3-9)
Corollary 3.1.6 Suppose that	is a basis for X. Then	and
the coefficient functionals	constitute a biorthogonal system.
We leave the proof to the reader (Exercise 3.3). For completeness we mention the following results about the coefficient functionals; they are proved in e.g., [210], [279].
Theorem 3.1.7 Let	be a basis for X and let {cfc}^2=1 be the
associated coefficient functionals. Then
(i) {cfc}^-! is a basis for its closed span in X*, and its associated biorthogonal system is	(considered as elements in X**).
(ii) If X is reflexive, then {cfc}^ is a basis for X*.
3.2 Bessel sequences in Hilbert spaces
The rest of this chapter concerns sequences in Hilbert spaces (see our conventions stated on page 46). For convenience we index all sequences by the natural numbers in this section. We shall soon see that all results actually hold with arbitrary countable index sets.
3.2 Bessel sequences in Hilbert spaces
51
Lemma 3.2.1 Let {fk}^-! be a sequence in H, and suppose that 22^ Ckfk is convergent for all {q}^ e ^2(N). Then
T : £2(N) -> H, T{ck}^=l := f\kfk	(3.10)
k=l
defines a bounded linear operator. The adjoint operator is given by
T*:H-^2(N),	=	(3.11)
Furthermore,
Ек/. ля2 < imi2 ii/ii2, v/ен. (3.12) k=l
Proof. Consider the sequence of bounded linear operators
Tn : £2(N) К Tn{ck}^ := jfckfk.
k=l
Clearly Tn —> T pointwise as n -> сю, so T is bounded by Theorem A.5.1.
In order to find the expression for T*, let f G 7^,	G ^2(N). Then
= </,ECk/fc)H = E(/,a)q.	(3.13)
k-1	k=l
We mention two ways to find T*f from here.
1) The convergence of the series	fk)ck for all	C ^2(N)
implies that {(/, fk)}^} G ^2(N); see for example [174], page 145. Thus we can write
(f,T{ck}f=1)H = ({(/,A)},{Q})^N)
and conclude that
r*/ = {(/,A)}£i-
2) Alternatively, when T : ^2(N) -> TL is bounded we already know that T* is a bounded operator from TL to ^2(N). Therefore the /с-th coordinate function is bounded from TL to C; by Riesz’ representation theorem, T* therefore has the form
r*/ = {(/,Pfc)}r=i
for some {gk}^-i in TL. By definition of T*, (3.13) now shows that
E</,= E(/,	e £2(N), f e H.
k=l	k=l
It follows from here that gk = fк-
52
3. Bases
The adjoint of a bounded operator T is itself bounded, and ||T|| = ||T*11.
Under the assumption in Lemma 3.2.1, we therefore have
IM2 < 11Л12 ll/ll2, v/ен,
which leads to (3.12).	□
Sequences {fk}kLi for which an inequality of the type (3.12) holds will play a crucial role in the sequel.
Definition 3.2.2 A sequence {fkYjfLi H is called a Bessel sequence if there exists a constant В > 0 such that
oo ii/ii2,	(3.14)
Every number В satisfying (3.14) is called a Bessel bound for {fk}kLi-
Theorem 3.2.3 Let {fk}^-i be a sequence in H. Then {fk}fc=i is a Bessel sequence with Bessel bound В if and only if
oo
T . {ck}T=l ^Ckfk
is a well-defined bounded operator from ^2(N) into H and ||T|| < л/В.
Proof. First assume that {fk}fc=i is a Bessel sequence with Bessel bound B. Let	G ^2(N). First we want to show that	is well-
defined, i.e., that Ckfk is convergent. Consider n,m 6 N, n > m. Then
n	m
k=l	k=l
n
E
fc=m+l
ckfk
= sup ( V ckfk,g} 11^11=1 k-m+l
n
<	sup V Mfk,g}\
llsll=« k=m+l
In \M2	/ n	\V2
<	( E №) sup ( e k/*»p>i2 I
\k=m+l z	llflll-1 \fc=m+l	Z
/ n \M2
<	Vb £ N2) 
Since	G £2(N), we know that	is a Cauchy se-
quence in C. The above calculation now shows that {E)fc=i ckfk}<^—1 is a
3.2 Bessel sequences in Hilbert spaces 53
Cauchy sequence in H, and therefore convergent. Thus T{q}^=1 is well-defined. Clearly T is linear; since ||Г{С4£1П = зирц9ц=1	a
calculation as above shows that T is bounded and that |\T\| < >/B. For the opposite implication, suppose that T is well-defined and that ||T|| < \fB. Then (3.12) shows that {fk}^=i is a Bessel sequence with Bessel bound B.
□
Lemma 3.2.1 shows that if we only need to know that {fk}kLi is a Bessel sequence and the value for the Bessel bound is irrelevant, we can just check that the operator T is well defined:
Corollary 3.2.4 If {fk}k^i is a sequence in H and '£d°-1Ckfk is convergent for all {cfc}g=1 6 ^2(N), then {fk}^=1 is a Bessel sequence.
The Bessel condition (3.14) remains the same, regardless of how the elements {fk}kLi are numbered. This leads to a very important consequence of Theorem 3.2.3:
Corollary 3.2.5 If {fk}kLi is a Bessel sequence in H, then Ckfk converges unconditionally for all	6 ^2(N).
Thus a reordering of the elements in {fk}^=1 will not affect the series Y^kLi ckfk when {cfc}^ is reordered the same way: the series will converge towards the same element as before. For this reason we can choose an arbitrary indexing of the elements in the Bessel sequence; in particular it is not a restriction that we present all results with the natural numbers as index set. As we will see in the sequel, all orthonormal bases, Riesz bases, and frames are Bessel sequences.
It is enough to check the Bessel condition (3.14) on a dense subset of H:
Lemma 3.2.6 Suppose that {fk}kLi is a sequence of elements in H and that there exists a constant В > 0 such that
fl</,A>|2<B||<
k=l
for all f in a dense subset V ofH. Then {Д}^_1 is a Bessel sequence with bound B.
Proof. We have to prove that the Bessel condition is satisfied for all elements in H. Let g G 7Y, and suppose by contradiction that
oo
E|(s,A)I2>b||5||2.
Then there exists a finite set F C N such that
El<3,Л>12 > В 1Ы12.
54
3. Bases
Since V is dense in H, this implies that there exists h G V such that
£|<л,А)12>В|Н12-
keF
but this is a contradiction. We conclude that
oo
El(s,A)|2 <B llsll2, VgeH.	□
fc=l
See Exercise 3.7 for a result in the same spirit.
3.3 Bases and biorthogonal systems in 77
We now return to some of the concepts defined in Section 3.1. The first lemma actually holds in Banach spaces, but for our purpose it suffices to consider a Hilbert space 7Y. Note that 7Y* = H; thus, if a sequence {fk}kLi in H has a biorthogonal sequence {дк}™^ then also {дк}<^=1 is a sequence in H.
Lemma 3.3.1 Let {fk}kLi be a sequence in H. Then
(i) {fk}kLi has a biorthogonal sequence {gkYkLt if and only if {fk}kLi is minimal.
(ii) If a biorthogonal sequence for {fk}kLi exists, it is uniquely determined if and only if {fk}kLi complete in H.
Proof. Suppose that {fk}*^ has a biorthogonal system {gk}%Li- Then, for any given j £ N,
(fj,gj} = 1 and {fk,gj} = 0 for к # j.
Therefore fj £ зрап{Д}/с^;, i.e., {Д}^=1 is minimal. For the other implication in (i), assume that {fk}kLi is minimal, and let
Ho := spanf/fc}^.
Given j G N, let Pj denote the orthogonal projection of H onto span{/fc}fc?£j, and let Io be the identity operator on Ho- Then it follows that (Zo - Pj)fj 0 0, and
(/о - лю = (Pjfi + Vo - Pj)fva» - Pi)fj> = ii(/o - л)лн2 / о.
For к / j, clearly (A, (Fo - Pj^fj) = 0- By defining
.. (1. - P,>!_,	6 N
11(41-iMII2 1
we obtain that {gk}^-i is a biorthogonal system for (fk}k)=i-
3.3 Bases and biorthogonal systems in TL
55
For the proof of (ii), assume that {fk}kLi has a biorthogonal system If {Altii is not complete we can replace by +
for some hk € Hq \ {0} and hereby obtain a new biorthogonal system for {fk}^L1- On the other hand, we leave it to the reader to verify that if {/fc}j£=i complete, then the biorthogonality condition can at most be satisfied for one family {p/Jfcti •	D
Theorem 3.3.2 Assume that {e^}^ is a basis for the Hilbert space H. Then there exists a unique family {gk}^=i inH for which
oo
/ =	V/eH.	(3.15)
k=l
is a basis for H, and	and {gk}kLi are biorthogonal.
Proof. By Corollary 3.1.5 the coefficient functionals {q}^ associated to are continuous; using Riesz’ representation theorem A.6.3, there exists a unique family {^}^=1 in H such that
that is,
oo
/ = E(/,5fc)eb VfeH.
k=l
We leave it to the reader to verify that no other family {gk}^=i can satisfy (3.15) and that {e/J^ and {gk}^-i are biorthogonal. The fact that {dk}kLi is a basis for H follows from Theorem 3.1.7.	□
The basis {gkYk-! satisfying (3.15) is called the dual basis, or the biorthogonal basis, associated to	It is interesting to observe that
the Bessel condition on	implies some kind of “opposite inequali-
ties” for {gk}k>-Ti inequalities of this type will play an important role as soon as we have defined frames in Chapter 5.
Lemma 3.3.3 Let {cfc}^ be a basis for H and {gk}kLi the associated biorthogonal system. If {e^is a Bessel sequence with bound B, then
(i) i ll/ll2 < П°=! l</,Pfc>|2,
6'V i Iе*I2 < lEkii с*Рл||2 f°r aW finiie sequences .
56
3. Bases
Proof. Let / G H. Using / =	and Cauchy-Schwarz
inequality, we obtain that
ll/l|4 =
fc=l
< Eia,^)i2
Л = 1	fc = l
< wii2 Ek/^ci2-
k-l
(i)	follows from this. For the proof of (ii), let {qJ-jEj be a finite sequence. Using the biorthogonal system	we can write
{<*}£=! =
oo
* <Ec^’e*>
, j=i
and
Note that it is essential that {q}^ is finite in (ii); for general sequences {cfc}£2-i G -62(N), the series CkQk might not converge (Exercises 3.2).
3.4 Orthonormal bases
We are now ready to introduce one of the central themes, namely, orthonormal bases in Hilbert spaces. They are the abstract (infinite-dimensional) counterparts of the canonical bases in C71, and have many similar properties. Orthonormal bases are widely used in mathematics as well as physics, signal processing, and many other areas where one needs to represent functions in terms of bases.
Definition 3.4.1 A sequence	in H is an orthonormal system if
(Cfe, Gj) = &k,j •
An orthonormal basis is an orthonormal system	which is a basis
for H,
Note that an orthonormal system {e^}^ is a Bessel sequence. In fact, if G £2(N) and m,n G N,n > m, then
n	m
5П	c^k
к-l	k=l
У2 ckek
k-m+l
E Ы2;
3.4 Orthonormal bases
57
as in the proof of Theorem 3.2.3 this implies that and that
oo	2 oo
^ckek = £>|2.
k — 1
Efcli ckek is convergent,
The next theorem gives equivalent conditions for an orthonormal system {efc}£Ti to be an orthonormal basis.
Theorem 3.4.2 For an orthonormal system	the following are
equivalent:
(i)	is an orthonormal basis.
(ii) f = 52/Xi (Л ek)ek, Vf EH .
(Щ) (f,g) = Y)T=i(f<ek)^k,g), 4f,geH.
(iv)	n°=il</>efc)l2 = ll/ll2, v/ен.
(v)	span{ek}kL1 = 'H.
(vi)	If (f, ek) = 0, Vfc 6 N, then f = 0.
Proof. For the proof of (i) => (ii), let f G H. If {ek}£ТХ is an orthonormal basis, there exist coefficients {q}^_1 such that f = 52i£=i ckek. Given any j E N, we have (/, ej) = 52£li ck^k,j = Cj, and (ii) follows, (iii) is an obvious consequence of (ii), and (iv) is a special case of (iii). The implications (iv) => (v) => (vi) are clear. For the proof of (vi)=> (i), let f EH. Since {efc}^_1 is a Bessel sequence we know that g := 52/Xi (A ek)ek is well defined; furthermore, (/ — g,ej) = 0 for all j 6 N, so by (vi), f = g = 52/Xi (Л ek)ek. To prove that {ед:}^_1 is a basis we only need to show that no other linear combination of {ekJ-JLj can be equal to /, and this follows by the argument we used to prove that (ii) follows from (i). □
The equality in (iv) is called Parseval’s equation. Via Corollary 3.2.5, we obtain the following important consequence of Theorem 3.4.2:
Corollary 3.4.3 If {efc}^ is an orthonormal basis, then each f EH has an unconditionally convergent expansion
oo
f = ^f’e^-	(316)
fc=l
In particular, the dual basis equals the basis itself.
58
3. Bases
Theorem 3.4.4 Every separable HUbert space H has an orthonormal basis.
Proof. Since H is assumed separable, we can choose a sequence {fk}^i in H such that spah{fk}(^L1 = H. By extracting a subsequence if necessary, we can assume that for each n G N, fn+i £ sPan{A}fc=i- By applying the Gram-Schmidt process to {fk}k^i we obtain an orthonormal system {ek}kLi in H for which spaUfe/J^ =	= H.	□
Often we want to have a concrete orthonormal basis for a given Hilbert space, rather than just its existence. The simplest case is ^2(N):
Example 3.4.5 Let be the sequence in ^2(N) whose /с-th entry is 1, and all other entries are zero. Then	is an orthonormal basis for
£2(N); it is called the canonical orthonormal basis. We will often denote this special basis by	□
We will later construct orthonormal bases for other Hilbert spaces, e.g., L2(-7r,7r) and L2(R).
Orthonormal bases are certainly the most convenient bases to use because the biorthogonal basis equals the basis itself. That is, the representation (3.16) is directly available, while the representation (3.15) via a general basis requires that we find the biorthogonal sequence	Unfortunately
we pay a price for this nice property. In fact, the conditions for {е^}^ being an orthonormal basis are strong, and often it is impossible to construct orthonormal bases satisfying extra conditions. We discuss this in more detail in Chapter 4. Note also that it is not always a good idea to use the Gram-Schmidt orthonormalization procedure to construct an orthonormal basis from a given basis: it might destroy special properties of the basis at hand. For example, the special structure of Gabor bases and wavelet bases (to be discussed later) will get lost.
Based on Theorem 3.4.4 we can prove that every separable Hilbert space can be identified with ^2(N):
Theorem 3.4.6 Every separable infinite-dimensional Hilbert space H is isometrically isomorphic to £2(N).
Proof. Let	be an orthonormal basis for H. We have already
observed that	tye* is convergent for all {q}^ G ^2(N). Further-
more each f G H has a unique expansion with ^-coefficients, namely / = 52(/, ek)ek. By letting	be the canonical orthonormal basis
for ^2(N), we can thus define the operator
c/(£Cfcefc) =£сл, {qK°=i e <*(N).
3.4 Orthonormal bases 59
Then U maps Я bijectively onto £2(N). For f G LL, f = ^2{f,ek)ek, we have
IIW =
= El(/.efc)|2
= ll/ll2;
thus U is an isometry.	□
The following theorem characterizes all orthonormal bases for LL starting with one orthonormal basis.
Theorem 3.4.7 Let {efc}^ be an orthonormal basis for Lt. Then the orthonormal bases for LL are precisely the sets {Uek}kL1, where U :Li -+LL is a unitary operator.
Proof. Let	be an orthonormal basis for LL. Define the operator
и : H H, U (£ckek) = ^ckfk. Ыы 6 ЛК).
Then U maps LL boundedly and bijectively onto LL. For /, g G LL, write f = ^{f,ek)ek and g = ^(g,ek)ek-, then, via the definition of U and Theorem 3.4.2,
(U*Uf,g) = {Uf,Ug}
= {'^{f>ek')fk,^/(g,ek}fk^
= ^2(f,ek}(g,ek) = (f,g);
thus U*U = I. Since U is surjective, it follows that U is unitary. On the other hand, if U is a given unitary operator, then
(Uek,Ue3) = {U*Uek,e3} = {ek,e3} = 6k,3,
i.e., {Uek}^Ll is an orthonormal system. That it is a basis follows from the fact that U is surjective.	□
Condition (iv) in Theorem 3.4.2 has an interpretation in terms of frames, see Definition 5.1.2. Without assuming that	is an orthonormal
system, it implies that	is an orthonormal basis if the vectors are
normalized:
Proposition 3.4.8 Assume that	is a sequence of normalized
vectors in Lt and that
oo
El(/,efc)|2 = Ц/Il2, V/67/. k=l
Then {ek}£Lj is an orthonormal basis for LL.
60
3. Bases
Proof. By Theorem 3.4.2 we only have to prove that {e/J^ is an orthonormal system. For each j G N we have
oo
1 = 1Ы12 = 52Kej>efc)|2 = 1 + ^2 |<e7,efc)|2,
k=l	k^j
which shows that (ej, efc) = 0 for к 0 j.
3.5 The Gram matrix
If {fk}kLi is a Bessel sequence we can compose the bounded operators T* and T; hereby we obtain the bounded operator
f / oo	\ V 30
T‘T;£2(N) ^2(N), T*T{ck}^ = 1	.
I \/=l	/ J fc=l
Letting	be the canonical orthonormal basis for ^2(N), the j/c-th
entry in the matrix representation for T*T is
(T*Tek,ej) = {Tek,Tej) =
Identifying T*T with its matrix representation, we write
T*T = {{fk, /ж=1.
The matrix {(A,is called the Gram matrix associated with {A}kLi, and the above argument shows that it defines a bounded operator on ^2(N) when {fk}k^i is a Bessel sequence. One can in principle consider the Gram matrix associated to any sequence {fk}kLi in H, but if we want it to define a bounded operator on ^2(N) we can not avoid the Bessel condition:
Lemma 3.5.1 For a sequence {fk}kL\ % following are equivalent:
(i) {fk}^i is a Bessel sequence with bound B.
(ii) The Gram matrix associated to {fk}kLi defines a bounded operator on ^2(N); with norm at most B.
Proof. The implication (i) => (ii) follows from the arguments above together with the norm estimate ||T|| < л/В in Theorem 3.2.3. Now assume that (ii) is satisfied, and let {c&Jjg-j G ^2(N). Then
X	<в2^\ск\2.
j=l k = l	k=l
(3.17)
3.5 The Gram matrix 61
Given arbitrary n, m e N, n > m,
E ck h - E Ckfk fc=i	*=i
E ck{fk,fj) k—m+l
where Cauchy-Schwarz’ inequality was used on the sum over j in the last step. Via (3.17) applied to the finite sequence
(' * , 0,0, cm.|-i, cm_|_2j *   , cn, 0,0, • • •),
k=m+l
j=m+l
oo n
< E E
k=m+l
В2 E k,l“.
j=m+l
Altogether we arrive at
n	m
^Ckfk ~ У^Ск/к
k=l	fc=l
It follows that Ckfk is convergent and, by repeating the argument,
oo
Ec^
k — 1
By Theorem 3.2.3 we conclude that {А}£1 is a Bessel sequence with bound B.	□
Lemma 3.5.2 Assume that {fk}kLi is a Bessel sequence in TL with preframe operator T. Then the Gram matrix defines an injective operator from Нт* into Нт*  Its range is dense in Нт* •
62
3. Bases
Proof. It is clear that T*T maps into itself. This restriction of T*T is injective: if {q}^ G and T*T{cfc}^=1 = 0, then
1|ГЫГ=1Н2 = (T*T{ck}^ {q}^) = 0,
i.e., {q}^ G Clfi/т = {0}- Using that H = 11т +Л/т*, we see that тгт* = T*H = T*7£?,
so 1Zt*t is dense in T^t* by continuity of T*.	□
Proposition 3.5.4 will give a sufficient condition for {fk}kLi being a
Bessel sequence. The proof uses Schur’s Lemma:
Lemma 3.5.3 Let M = {М3^}^к^1 a ma^x for which Mjtk = Mktj for all fi к G N, and for which there exists a constant В > 0 such that
oo
VjeN.
fc=l
Then M defines a bounded operator on ^2(N) of norm at most B.
Proof. Let	G ^2(N). The assumptions imply that M{ck}^L{
is well defined as a sequence indexed by N, whose j-th coordinate is Mjikck. It is, however, not immediately clear that this sequence belongs to ^2(N). Abusing the notation, it is enough to show that the map
-> {{с1к}^ъМ{ск}^=1}^	(3.18)
is a continuous linear functional on ^2(N). In fact, this implies that belongs to the dual of ^2(N), which is ^2(N) itself. Now, for
{<4}^! e £2(N),
52 52 -^j,kckdj
Using Cauchy-Schwarz’ inequality, (\ 1/2 /	\ 1/2
EEim^iic.i2]	к j
j=i fc=i	/ \j=i	у
/ oo \ 1/2 / oo \ 1/2
< в £ы2	52ki2	•
\/c = l	/	\j = l	/
3.6 Riesz bases 63
This shows that (3.18) indeed defines a continuous linear functional on /2(N), so M maps ^2(N) into ^2(N). Also,
||M{Cfc}^=1|| = sup |{{dfc}r=l>M{Cfc}£l)^(N)| ll{<MII=l
/со	\ V2
< В	>
\fc=l	/
which completes the proof (see Exercise 3.11 for a question about the proof).	□
An application of Schur’s lemma gives a sufficient condition for the Gram matrix defining a bounded operator on ^2(N), and thus for {A}^ being a Bessel sequence. For the proof we just have to refer to Lemma 3.5.1:
Proposition 3.5.4 Let {fk}kLi be a sequence in H and assume that there exists a constant В > 0 such that
oo
k=l
Then {A}£Li is a Bessel sequence with bound B.
Compared with the Bessel condition (3.14), Proposition 3.5.4 has the advantage that it only involves inner products between the elements in {fk}kLi', that is, only a countable number of conditions must be verified, while the Bessel condition has to be checked for all f G H.
3.6 Riesz bases
In Theorem 3.4.7 we characterized all orthonormal bases in terms of unitary operators acting on a single orthonormal basis. Formally, the definition of a Riesz basis appears by weakening the condition on the operator:
Definition 3.6.1 A Riesz basis for В is a family of the form {Uek}^L1} where	is an orthonormal basis for В and U :B -» H is a bounded
bijective operator.
A Riesz basis is actually a basis (Exercise 3.6). In fact, one can characterize Riesz bases in terms of bases satisfying extra conditions:
Lemma 3.6.2 A sequence {fk}kLi is a Riesz basis for В if and only if it is an unconditional basis for В and
о < inf ll/kll < sup ||/*|| < oo.
к	k
64
3. Bases
Lemma 3.6.2 was proved by Kothe and Lorch and has been rediscov-ered/reproved many times (see the discussion in [279]). We refer to e.g., [145] or [210] for a proof.
The dual basis associated to a Riesz basis is also a Riesz basis:
Theorem 3.6.3 If {fk}kLi is a Riesz basis for H, there exists a unique sequence {gk}^=1 in H such that
oo
/ = E</,^)A, v/ew.	(3.19)
k=l
{gk}<^-1 is also a Riesz basis, and {fk}kLi and {gk}^-! are biorthogonal. Moreover, the series (3.19) converges unconditionally for all f Z'H.
Proof. According to the definition we can write
where U is a bounded bijective operator and {e^ is an orthonormal basis. Let now f E H. By expanding U-1/ in the orthonormal basis , we have
oo	oo
U^f = Y^f,ek)ek = k=l	k=l
Therefore, with gk := (C/-1)*ek,
oo
f = UU~1f = ^{fAU-^e^Ue, k—1 k—1
Since (L-1)* is bounded and bijective, {gkYj^i is a Riesz basis by definition. For f E H,
oo	oo
Ek/,a)i2 = Ei</,tW = Hir/ii2 fc=l	fc=l
< F*ll21Ш12
= ||t/||2 11/lP,	(3.20)
this proves that a Riesz basis is a Bessel sequence. Thus, the series (3.19) converges unconditionally by Corollary 3.2.5 (this was stated without proof in Lemma 3.6.2). The rest follows from Theorem 3.3.2 (or direct verification).	□
We call {gkY^-t the dual Riesz basis of {А}£Тр We can of course apply Theorem 3.6.3 to find the dual of {gkYk-i = {(^’“1)*efc}fci1; by the proof of Theorem 3.6.3 we have to find the adjoint of the inverse of (t/-1)*
3.6 Riesz bases 65
and apply result to {efc}^. This procedure gives us {fkYjfi=1 back. Therefore {fk}^Li and {gk}^=1 are duals of each other, and
oo	oo
f = ^(f,9k)fk = RfA, V/ e я.	(3.21)
k=l	k—1
For later use we note that a Riesz basis is a Bessel sequence and also satisfies some kind of “opposite inequality”:
proposition 3.6.4 If {fk}^=1 = {Uek}<^=1 is a Riesz basis forH, there exist constants А, В > 0 such that
oo
Л ll/ll2 < E K/-M < B ll/ll2, V/ 6 H. (3.22)
The largest possible value for the constant A is	and the smallest
possible value for В is ||[/||2.
Proof. That a Riesz basis {Uek}jfL} is a Bessel sequence with optimal upper bound ||17|| follows already from the estimate in (3.20). The result about the lower bound follows from
ll/ll = ll(t/*)-1t/*/ll < ll(t/*)’1ll ll^/ll = НС/"1!! P*/l|.	□
A standard way of constructing an operator is to define it on a basis and then extend by linearity. The lemma below gives some conditions for this being possible.
Lemma 3.6.5 Let H,JC be Hilbert spaces, and let {hk}^L1 be a sequence in H, {дк}^=1 a sequence in K,. Assume that	is a Bessel sequence with
bound B, that	is complete in H, and that there exists a constant
A > 0 such that
A ЕЫ2<||ЕсЛ||2	(3-23)
for all finite scalar sequences {q}. Then
u (Eq/i,£) := EQ5fc finite^
defines a linear bounded operator from span{hk}(^L1 into span^Y^ and U has a unique extension to a bounded operator from H into /С; the norm of U as well as its extension is at most
Proof. By the assumption (3.23), every h € span{/i/c}^.1 has a unique representation h = ^Ckhk with {ck} finite; it follows that U is well defined and linear. Given a finite sequence {c^},
66
3. Bases
IHMI’ = IIE-1I2
< в £ la |2
Thus U is bounded. Since {hkYjfi-t is assumed to be complete, U has an extension to a bounded operator on К The rest is standard.	□
The next theorem gives equivalent conditions for {fkYjfi-i being a Riesz basis. Note in particular condition (ii), which will be used throughout the book and, in fact, by several authors is used as the definition of a Riesz basis.
Theorem 3.6.6 For a sequence {fk}kLi in the following conditions are equivalent:
(i) {A}£i zs a Riesz basis for H.
(ii) {A}fcLi is complete in H, and there exist constants A,В > 0 such that for every finite scalar sequence {c>} one has
aEi^i2 ||E^a||2 -вЕы2- (3-24)
(Hi) {fk}kLi is complete, and its Gram matrix {(fk^fj)}^^ defines a bounded, invertible operator on £2(N).
(iv) {fk}^=1 is a complete Bessel sequence, and it has a complete biorthogonal sequence	which is also a Bessel sequence.
Proof. (i)=>(ii). Assume that {fk}kLi is a Riesz basis, and write it in the form {Uek}(^L1 as in the definition. Note that {fk}kLi is complete. Given any finite scalar sequence {q},
||Ec^||2 = IIе7 (Ес^)|Г n< ||EcH|2 = ||г||Ты2
and
||E4f=Ik’1'' к-т11ЕЧГ
from which we deduce that
EK < ||Е«л|Г я*Еы’
(ii)=>(i). The right-hand inequality in (3.24) implies that {fk}(^=1 is a Bessel sequence with bound В (Exercise 3.9). Choose an orthonormal basis for Fl, and extend by Lemma 3.6.5 the mapping Uek := fk to a
3.6 Riesz bases
67
bounded operator on H. In the same way, extend Vfk * = ek to a bounded operator on Then VU = UV = I, so U is invertible; thus {fkYkLi is a Riesz basis.
(i)=>(iii). Write again {fk}k>=1 = {Uek}kLv For any k,j e N,
(А,Л) = (^,[/е5) = ([/*^,е5)
i.e., the Gram matrix is the matrix representing the bounded invertible operator U*U in the basis {ек}£г
(iii)=>(ii)• Assume that (iii) is satisfied. Then Lemma 3.5.1 together with Theorem 3.2.3 shows that the upper condition in (3.24) is satisfied. Let G denote the operator on ^2(N) given by the Gram matrix {(А, Л)}^=1 • Given a sequence	e ^2(N), the j-th element in the image sequence
G№ is	Thus
OO oo
<G{ck}r=1,{ck}?=l) =
j=l k=l
ckfk k=l
Thus G is positive, and a similar calculation shows that G is self-adjoint. Let V denote the square-root of G, cf. Lemma A.6.7. Then the above calculation gives that
oo
ckfk
k=l
1 jpFT
oo
Еы2
k=l
2
(i) => (iv). Follows from Theorem 3.6.3 and Proposition 3.6.4.
(iv) => (i). Every f e span{A}£Li has a representation f = ^ckfk for a finite sequence {q}, and under the assumptions in (iv) it is unique: if f — 'ESkfk, then ck = (f,gk)- Letting {е^}^ be an orthonormal basis for 7Y, we can therefore define an operator
V : span{A}£°=1 -> К Vfk = ek.
Writing f e span{A}/Xi as f = ^{f,gk)fk, and letting C denote a Bessel bound for {gk}k>-!, we have
nW = ||Е</>^Н|2 = EK/rf < СЦ/Ц2.
By completeness of {A}£Lp V has an extension to a bounded operator on %. Since the assumptions in (iv) are symmetric in A and gk we can also extend Tgk '•= ek to a bounded operator on H.
68
3. Bases
Consider finite linear combinations of {fk}kLi and {«ZfcJSXi» 8аУ»
f = ^ckfk, g = '^dk9k-
Because {A}/Xi and {gk}^=1 are biorthogonal, we have
(Vf,Tg) = (^скек,^2<1кек^ = ^ckdk =
by continuity and completeness we therefore have (Vf,Tg) = (f,g) for all Thus, for any h E H,
\\h\\2 = {h,h} = {Vh,Th)<\\vh\\ НГЦ ЦЛЦ.
It follows that V is injective. V is also surjective: Given g E H, write g =	= V	• Since fk = V-'ek, we conclude'
that	is a Riesz basis.	□
A sequence {A}£Li satisfying (3.24) for all finite sequences {q}^ is called a Riesz sequence. By Theorem 3.6.6 a Riesz sequence {A}£Li is a Riesz basis for span{A}/£Li, which might just be a subspace of H. Note that if the condition (3.24) is satisfied for a family {fk}^=i, then it is clearly satisfied for any subsequence of {A}£i • This leads to the following important consequence of Theorem 3.6.6.
Corollary 3.6.7 Every subfamily of a Riesz basis is a Riesz sequence.
If (3.24) holds for all finite scalar sequences {q}, then it automatically holds for all {ck}<^=1 € ^2(N) (Exercise 3.9). If {A}^ is a Riesz basis, numbers А, В > 0 which satisfy (3.24) are called lower Riesz bounds, respectively, upper Riesz bounds. They are clearly not unique, and we define the optimal Riesz bounds as the largest possible value for A and the smallest possible value for B. The optimal Riesz bounds can be characterized in terms of the operators appearing in the proof of Theorem 3.6.6:
Proposition 3.6.8 Let {fk}kLi = {Uek}(^=1 be a Riesz basis for H, and let G : £2(N) —> £2(N) be the Gram matrix. Then the optimal Riesz bounds are
Proof. The bounds involving U follow directly from the proof of Theorem 3.6.6. Also, by Lemma A.6.1,
Ill’ll = \\U*U\\ = ||!7||2 and ||G’-1|| = \\(U*U)~l\\ = |К||2.
That the optimal upper Riesz bound equals ||G|| was also proved in Lemma 3.5.1.	□
3.7 Fourier series and Gabor bases 69
Note that the same optimal bounds involving U were obtained in the inequalities in Proposition 3.6.4.
If (3.24) holds with A = В = 1, the sequence {fk}^i is orthonormal:
proposition 3.6.9 Assume that span{fk}(^_1 = H and that
for all finite scalar sequences {q}. Then {А}£1 is an orthonormal basis for H.
proof. The assumptions imply by Theorem 3.6.6 that {A}jg=i is a Riesz basis for H, so by letting	be an orthonormal basis for H we can
write {fk}™-! = {UekYkL} for an appropriate bounded invertible operator U. Then, for all {ck}^=1 E £2(N),
oc
Еы2 =
к—1
Ckfk к—1
(oo
Ес№
к=1
2
It follows from here that ||C7|| = \\U J|| = 1; by Proposition 3.6.4 we conclude that
oo
El(/,A)l2 = ll/ll2, V/6?/.
fc=l
Since ||A|| = 1, Vfc E N, we now obtain the result via Proposition 3.4.8. □
We have in this Section concentrated on theoretical properties of Riesz bases in general Hilbert spaces; we will return to more concrete results about Riesz bases in e.g., L2(K) later. For now we just mention one class of Riesz sequences, which we will use in Chapter 14:
Lemma 3.6.10 Let n E N and consider the B-spline Bn defined in Section A.9. Then {Bn(- — k)}kez is a Riesz sequence in L2(R).
The result is clear for n = 1 because {Bi(- — k)}ke% Is an orthonormal system. For n > 1 it is an easy consequence of Theorem 7.2.3, so we postpone the proof till page 146.
3.7 Fourier series and Gabor bases
Let us now consider some concrete orthonormal bases for the function spaces L2(0,1) and L2(IR). Here we will use other index sets than the natural numbers; as we have seen in Corollary 3.2.5, Bessel sequences can be
70
3. Bases
ordered any way we want without affecting the convergence of the relevant series expansions, so we can apply all results presented so far without problems.
The starting point is Fourier series. We expect the reader to be familiar with the basic theory, so we only give a short repetition.
Fourier series can be associated to functions in any space L2(Z), where I is a bounded interval in JR. For our purpose it will be convenient to consider functions in L2(0,1/d), where b > 0. Since the functions
ek(x) := b^2Ekb(x) = b^2e2lrikbx, UZ	(3.25)
constitute an orthonormal basis for L2(0, 1/6), every f € L2(0,1/b) has an expansion
f = Y^ckek,	(3.26)
where
fl/b
ck = (f, ek} = Ekb) = &V2 / f^e-2.ikb.d^ (3 27) Jo
The expansion (3.26) is called the Fourier series of /, and the numbers {ck}kez are the Fourier coefficients.
Note that with the above definition of Fourier series, the constant 61/2 appears in the expression for ek as well as in the coefficient ck. Often it is more convenient to express f E L2(0,1/6) as a linear combination of the functions е27ггА:Ь:с rather than ek; doing so, the Fourier expansion takes the form
ж	fl/b
f(x) cke2nikbx, where ck = b f(x)e~2™kbxdx. (3.28)
One has to be careful about which format one uses, especially when using different sources: some authors choose to call the coefficients ck for Fourier coefficients rather than ck. In order to avoid confusion we will usually simply speak about a Fourier expansion, and specify whether it is with respect to the functions ek or е2пгкЬх. We note that an advantage of using ek is that {e/c}^ is an orthonormal basis; another advantage is that the expansion (3.26) does not refer to the variable, and thus avoids the confusion with pointwise convergence (see below).
It is crucial to understand the exact meaning of the Fourier expansion (3.26). In the following discussion we take 6 = 1 for notational convenience. That is, we consider L2(0,1) and the orthonormal basis {е^}^, where ek(x) = e2lrlkx. The Fourier series of f E L2(0,1) is
/ =	where ck = f f(x)e~27rlkxdx.
3.7 Fourier series and Gabor bases
71
^ow the meaning of (3.26) is that
n
f- ECk&k
k— — n
W.1)
/ /(*) - E cke2nkx
V°	k= — n
—> 0 as n -> oo.
Convergence in L2(0, l)-sense is radically different from pointwise convergence, so we can not claim that f(x) =	for % £ [0,1]
without extra assumptions. The question of pointwise convergence of Fourier series is a very delicate issue (especially for functions not belonging to L2(0,1): there exist functions in 221 (0,1) for which the Fourier series diverges almost everywhere, despite the fact that the Fourier coefficients are well defined). As a positive result, the Fourier series for an arbitrary function in L2(0,1) converges pointwise almost everywhere. Furthermore, if f is a 1-periodic continuous and piecewise differentiable function, then
/(z) = ^{fiekjektx), Ух G [0,1].
fcGZ
If the derivative /' € L2(0,1), then for all N E N,
/w - E (f’e^ek(x)
\k\<N
1	1 / Г1 \ V2
• (3-29)
Via (3.29) we can estimate the number of terms we have to keep in the partial sum of the Fourier series in order to obtain a given pointwise approximation of
Let us return to Fourier series in L2 (0,1/5). We state a lemma, which is an immediate consequence of the functions	in (3.25) being an
orthonormal basis for L2(0,1/5).
Lemma 3.7.1 Let fig E L2(0,l/5) for some 5 > 0, and consider the Fourier series
f = ECfcefc) 9 = 'F;dkek,
with ek given by (3.25). Then
(f,g) = ^ckdk.
k£“Z
A function f which is a finite linear combination of exponentials,
72
3. Bases
is called a trigonometric polynomial. For a trigonometric polynomial, the Fourier series equals the function itself everywhere. Via Euler’s formulas
gQirikx । е~2тггкх	^ттгкх _ 2irikx
cos(2tt/cz) =--------------------, sin(27rA;a:) =----------—---------,
we see that any finite linear combination of functions sin(27rA;:r), cos(2tt/cz) is a trigonometric polynomial.
In the following example we show how to construct an orthonormal basis for L2(K) based on the orthonormal basis {e2lrlkx}kez for L2(0,1).
Example 3.7.2 Let X[o,i] denote the indicator function for the interval [0,1]. Then {e27rlfc:cX[o,i](^)}A:GZ is an orthonormal basis for L2(0,1); by translation we see that for each n G Z the space L2(n,n + 1) has the orthonormal basis {e27rlfc^-n^[o,i](^ - n)}fceZ = {е27ггЬ:X[o,i]“ n)}kez-Putting these bases together we obtain that L2(K) has the orthonormal basis
Note that all elements in the basis consist of translated versions of X[o,i] which have been modulated, i.e., multiplied with a complex exponential function; using the operators introduced in Section 2.5 we can write the basis as {EkTng}kine%, where g = X[o,i]• Bases of the form {EkTng}kinez are called Gabor bases. Calculations with Gabor bases are convenient because of their coherent structure, i.e., the fact that the elements in the basis appear by the action of a family of operators, namely EkTn,k,n G Z, on the single function g. We will consider some of the limitations on such bases in Chapter 4 and extensions to frames in Chapters 8-9.	□
3.8 Wavelet bases
Wavelet bases constitute another important class of bases. Given a function G L2(K) and j, к G Z, let
^3,k(x) := V^Vx -k), x G R	(3.30)
In terms of the translation operators Tk and the dilation operator D introduced in Section 2.5,
= DjTk<ilj, j,k G Z.
If	is an orthonormal basis for L2(JR), the function is called
a wavelet. The first example of such a function appeared long time before the systematic study of wavelet bases began around 1985:
3.8 Wavelet bases 73
Example 3.8.1 The Haar function is defined by ( 1	if 0 < x <
гр(х) = < —1 if | < x < 1,	(3.31)
[ 0 otherwise.
Already in 1910 it was proved by Haar [161] that the functions constitute an orthonormal basis for L2(K) for this choice of гр. For the orthonormality one can argue as follows. If we first consider гр^ь and гр^н, i.e., elements with the same dilation parameter, then
= {ОЗТк-ф^Тк'ф} = {Тк^,Тк.ф) = 6k,k’.
Now assume that j' / j, say, j’ > j. The commutator relations (2.11) give that
= (D3Tk^,D3'Tk^)
= (T-k.D3-3'Tk^^)
= (D3~3' T_k,v_,. +кф,чр}.
The function	+k^ has support in the interval
I : =	+/c),2j''-V^2j-/ 4- к 4- 1)[
= [-V 4-	-k' 4- 2^'~\к 4- 1)[.
The length of I is 2J “J, which can take the values 2,4,8,  Now, the support of гр has length 1, and is contained in an interval on which
Т_к,2з-31 +кгр is constant (make a picture!); it follows that
= У (PJ_/T_fc,2J-/+fcV') (x)-ip(x)dx = 0.
For the proof of the basis property we refer to [106], [173], or [288]. □
Stromberg [270] constructed in 1982 (before the wavelet era began) wavelet orthonormal bases {гр^к}з,ке% for which гр has exponential decay and гр E Cfc(IR); here к E N is arbitrary but fixed. Meyer [222], [206] found in 1985 wavelet bases for which гр E С°°(К) and гр E Cfc(IR), к E N. In 1986 Mallat and Meyer introduced multiresolution analysis as a general tool to construct wavelet orthonormal bases:
Definition 3.8.2 A multiresolution analysis for L2(K) consists of a sequence of closed subspaces {Vjjjez of L2(R) and a function ф e Vo, such that
(i)	• • • V_! C Vo C Vi • • •.
(ii)	UjVj = L2(R) and C\3Vj = {0}.
74
3. Bases
(iii)	f e Vg О [ж -> f(2x)] e vj+1.
(iv)	f ev0^Tkf e Vo, v/c e z.
(v)	{Ткф}ке2 is an orthonormal basis for Vo-
When (i) is satisfied we say that the spaces V3 are nested. This is a very convenient property in for example approximation theory, especially when there is an easy recipe for moving around between the spaces Vj. The later property is also guaranteed by Definition 3.8.2 because (iii) implies that Vj = DWq, i.e., that all of the spaces Vj are scaled versions of Vo-
lf we want to approximate a function f € L2(K) via a multiresolution analysis, the natural starting point is to search for an approximation within a certain V3-space. In case no element in this space approximates f well enough, we choose a larger j-value; then we obtain a better approximation, and it is taken from a space which is just a scaled version of the previous space.
Definition 3.8.2 is central in numerous constructions of orthonormal bases, and the topic is already well covered with many excellent books (see for example [139] and [286] for elementary treatments, or [106], [223], [288] for more advanced presentations). For this reason we will only describe how a multiresolution analysis can be used to construct an orthonormal basis.
Assume that the conditions in Definition 3.8.2 are satisfied. For j 6 Z, we let Wj denote the orthogonal complement of Vj in Vj+i. By letting Qj denote the orthogonal projection onto Wj, it follows from (i) and (ii) that each f e L2(R) has a representation f = Z^ezQ?/’ where Qjf-LQj>f for j j'] that is,
L2(K)=^Wj.	(3.32)
The spaces W3 satisfy the same dilation relationship as Vj, i.e.,
IVo H € Wj.	(3.33)
In order to obtain an orthonormal basis {^j,k}j,kez for L2(K) it is now enough to find ф € Wq such that {^(- - k)}kez is an orthonormal basis for Wo; via the dilation property (3.33) and (3.32) this implies that {'ipj,k}j,kez is an orthonormal basis for L2(K). One way of choosing ф is as follows. First, the condition ф G Vo C VJ implies by (iii) that
G Vo-
Since {?кФ}кех is an orthonormal basis for Vo, there exist coefficients {ojfcez C £2(Z) such that
ф=р-1ф = ^скТкф.
v2 kez
3.8 Wavelet bases
75
Using the Fourier transform and the commutator relations in (2.12) it follows that	= ^к^СкЕ-кФ', defining the 1-periodic function
:= QELfc, this can be written as
0(27) = H0(7)<^(7), a.e.7 e R.
One choice for a function ф E Wq generating a wavelet orthonormal basis for L2(R) is now given by
Ш=Я0(^ + 1)е-^(^).	(3.34)
Note the indirect definition of ф in terms of its Fourier transform. As we will see in Chapter 13 this is typical for wavelet constructions. We also mention that Corollary 13.4.8 will give a proof of the fact that the function ф in (3.34) generates an orthonormal basis; however, the more general approach in Chapter 13 is certainly not the most natural choice if one is only interested in the orthonormal case.
Since ф E Wq С Vj, there are coefficients {ckjkez € ^2(Z) such that
=	(3.35)
The coefficients {ckjkez can be found from (3.34) via manipulations on the Fourier coefficients of Hq, but we do not need them for this short description.
The Haar basis can be constructed via the multiresolution analysis defined by ф = X[o,i[ and
Vj = {f E L2(R) : f is constant on [2~JA;, 2~^(k + 1)], V/c E Z}.
In terms of the function ф, the Haar function in (3.31) is
Ф = ^^i,o "	(3-36)
The Haar function is a special case of a spline wavelet. In fact, one can consider higher order splines Bn (see Section A.9 for the definition) and define associated multiresolution analyses, which leads to wavelets of the type
ф(х)=^,скВп(2х-к).	(3.37)
fcez
These wavelets are called Battle-Lemarie wavelets. The coefficients {ckjk^z are calculated in e.g., [106]; except for the case n = 1, all coefficients q are non-zero, which implies that ф has support equal to R. However, the wavelets have exponential decay.
Mathematicians as well as engineers immediately recognized the importance of multiresolution analysis, and an intense research aimed at construction of desirable bases followed. Most of the important wavelet
76
3. Bases
bases for L2(JR) are constructed via the approach sketched above, e.g., the bases by Daubechies [106], where has compact support and is к times differentiable, к G N. However, not all wavelets can be constructed via multiresolution analysis. It is shown in e.g., [173] that ф € L2(JR) is a wavelet if and only if \\ф\\ = 1 and the equations
£|^(2^)|2 = 1,	(3.38)
(2^(7 4- q)) = 0 for all odd integers q (3.39) j=o
hold for almost all 7 e JR. Among all wavelets, the wavelets generated from a multiresolution analysis are characterized by the equation
00
££Й2>(7 + fc)|2 = 1, j=ikez
a result which is also proved in [173].
In Chapter 4 we point out some of the limitations on what can be obtained by wavelet bases. Later, Chapters 11-14 deal with frames having the wavelet structure.
3.9 Exercises
3.1	Prove that ||| • ||| (introduced in the proof of Theorem 3.1.4) defines a norm on X, and that X is a Banach space with respect to this norm.
3.2	Let {e/c}^! be an orthonormal basis for a Hilbert space H, and define	by fk = %ек, к € N.
(i)	Prove that {fk}kLi is a basis for H, and find the biorthogonal system {gk}^=1-
(ii)	Prove that the coefficient functionals associated to	are
not uniformly bounded.
(iii)	Show that there exists	€ ^2(N) for which 22^ Ckgk is
divergent.
3.3	Prove Corollary 3.1.6.
3.4	Let {efc}^_1 be an orthonormal basis for a Hilbert space 7/.
(i)	Prove that	u {e/J&i is linearly independent, but
not cxj-independent.
3.9 Exercises 77
(ii)	Prove that {ei} U {e^ 4- e^i}^ is ^-independent, but not minimal. (Hint: In Example 5.4.6 we prove that {e^ 4- efc+i}^ is complete).
3.5	Assume that {fk}^! is a Bessel sequence with bound B. Prove that (i) ПАП2 < В for all к e N;
(ii)	if ЦАЦ2 = В for some к 6 N, then fk-Lfj for all j E N \ {к}.
3.6	Prove directly via the definition that a Riesz basis is a basis.
3.7	Prove that if {A}^i is a sequence in a Hilbert space H and
oo
El(/,A)I2<^ W, fc=l
then {fk}kLi is a Bessel sequence.
3.8	Prove that the upper and lower conditions in (3.24) are unrelated: there exists a sequence {fk}%Li satisfying the upper condition for all finite sequences {ck}^=1, but not the lower condition; and vice versa.
3.9	Let	be a sequence in a Hilbert space H. Prove that
(i)	If there exists В > 0 such that
||Ec*a||
for all finite sequences {q}, then 52^ Ckfk converges for all {ca:}^_1 e ^2(N) and {fk}^! is a Bessel sequence with bound B.
(ii)	If (3.24) holds for all finite scalar sequences {q}, then it holds for all {ck}^ e £2(N).
(iii)	if {A}£i is a Riesz basis, then
^Ckfk is convergent о {ctJgtj 6 f2(N). k=l
3.10	Prove that a basis in a Hilbert space is minimal.
3.11	Consider the proof of Lemma 3.5.3. Where is the assumption
= Mk,j
used?
4
Bases and their Limitations
The next chapters will deal with generalizations of the basis concept, so it is natural to ask why they are needed. Bases exist in all separable Hilbert spaces and in practically all Banach spaces of interest, so why do we have to search for generalizations?
In this chapter we will give some answers to this question. As we will see, the main point is the missing flexibility: the conditions for being a basis are so strong that
•	it is often impossible to construct bases with special properties;
•	even a slight modification of a basis might destroy the basis property.
The starting point for a more detailed discussion must be to clarify why we are at all interested in bases! One reason is that a basis {e^} for a normed vector space X allows us to represent every f 6 X as a (maybe infinite) linear combination of the basis elements,
f = '^скек,	(4.1)
with coefficients {q} which depend linearly on f. We will refer to this by saying that {efc}^ has the expansion property. This property makes it possible to reduce many questions about elements in X to the elements {efc} in the basis. For example, the action of a bounded operator U on f can be found if we know the representation (4.1) and the action of U on the basis {e&}:
Uf=u(^cke^ = ^ckUek.
80
4. Bases and their Limitations
Bases are characterized by the expansion property (4.1) with unique coefficients {ck } associated to each f G X. One might ask whether uniqueness is really needed? Our answer is no: it is usually enough to know the existence of some usable coefficients, together with a recipe for finding them.
In this chapter we discuss some cases where (4.1) holds without {e^} being a basis. We begin with the simple observation that if {e^} is a basis for X and ф is an arbitrary element in X, then {e^} U {ф} is not a basis, despite the fact that each f G X has representations of the form
/ =	+ <1ф.	(4.2)
The reason is that {e^} U {ф} is no longer independent, i.e., several choices for the coefficients {c/J and d are possible: one choice is to take d = 0 and let {ck } be the coefficients representing f in the basis {e/J; another choice is to take {q} such that f — ф =	and d = 1.
By this argument, the basis property is destroyed when an arbitrary nonempty collection of vectors is added to {e^}, but the expansion property is preserved.
At first glance, the above construction might appear artificial: why would one like to add elements to a basis? One reason is that we gain some freedom: the coefficients in (4.1) are unique, but in (4.2) we can choose between several options. In Section 12.1 we will see a concrete case where this is very useful. Also, Section 5.9 will show that having more elements than needed for a basis has a certain noise suppressing effect.
Non-bases with the expansion property also appear naturally in function spaces:
Example 4.0.1 Let us return to the orthonormal basis {ekjkez for L2(0,1) considered in Section 3.7, i.e., the functions ek(x) = е21ггкх. Given an open subinterval I c]0,1[ with \I\ < 1, we can identify L2(I) with the subspace of L2(0,1) consisting of the functions which are zero on ]0,1[\Z. Hereby a function f 6 L2(I) is identified with a function (which we still denote f) in L2(0,1), and which has the expansion
/ = ^(/,efc)efcinL2(0,l).	(4.3)
к GZ
Since
1/2
f - 52
|fc|<n
2
L2(7)
/(x)- £ {f,ek)e™k* k — — n
/«- £ (/,efc)e2^ к— — n
—> 0 as n —> oo,
4. Bases and their Limitations 81
we also have
/ = £(/,efc)et in L2(I).	(4.4)
That is, the functions {ekjkez also have the expansion property in L2(I). However, they are not a basis for L2(I)\ To see this, define the function
f(x} _ / if x G 7, [ 1 if x I.
Then f € L2(0,1) and we have the representation
7 = E<7,e*)et in L2(0,1).	(4.5)
/cCZ
By restricting to 1, the expansion (4.5) is also valid in L2(I); since f = f on 7, this shows that
/ = E(7,efc)efc in L2(7).	(4.6)
kez
Thus, (4.4) and (4.6) are both expansions of f in L2(7), and they are non-identical; the argument is that since f ± f in L2(0,1), the expansions (4.3) and (4.5) shows that
{{f,ek)}kez # {(/, £k)}kez-
The conclusion is that the restriction of the functions {ekjkez to 7 is not a basis for L2(7), but the expansion property is preserved. In Example 5.4.5 we prove that {ekjkez is a frame for L2(7).	□
In a finite-dimensional vector space X we know that every family of vectors which spans X contains a basis (Exercise 1.1). In an infinitedimensional Hilbert space the situation is dramatically different: there exists a family of vectors {fk}kLi such that
•	each f € H has an unconditionally convergent expansion oo
/ = ЕС*Л with {q}£°=1 6 £2(N); fc=l
•	no subsequence of {fk}kLi is a basis for H.
We give a concrete construction in Section 6.4. Intuitively, this kind of example is difficult to understand: it shows that we might have the expansion property for families which have no relationship to a basis. This is also an argument for looking for generalizations of bases.
82
4. Bases and their Limitations
4.1 Gabor systems and the Balian-Low Theorem
In concrete Hilbert spaces like L2(K) we are able to give explicit constructions of bases, like the Gabor orthonormal basis
{e Х[0,1](ж ~ ^)}ттг,пЕ2 =	(^)}m,n£Z
in Example 3.7.2. However, exactly this example touches one of the limitations, as we will see now. Observe that
The fact that X[o,i] is discontinuous, and the oscillations and slow decay of X[o,i], makes the characteristic function unattractive from the point of view of e.g., time-frequency analysis. One could hope that better results could be obtained by replacing the function X[o,i] by a smoother function g; unfortunately, the Balian-Low Theorem shows that there are limitations on the properties g can have if we want {ЕтТпд}т>пЕ2 to be a Riesz basis:
Theorem 4.1.1 Let g € L2(JR). If {EmTng}minE% is a Riesz basis for L2(R), then
(J |тр(а;)|2^ (У |75(?)|2<Ь) = °0-	(4-7)
In words, the Balian-Low theorem means that a function g generating a Gabor Riesz basis can not be well localized in both time and frequency. For example, it is not possible that g and g satisfy estimates like
C	C
simultaneously. For proofs of the Balian-Low theorem we refer to [105], [20], or [173]. We note in passing that the Balian-Low theorem is close to describe the limit case of what can be obtained with Gabor bases. In fact, it has recently been proved by Benedetto et al. [17] that for any e > 0 we can construct orthonormal bases {ЕтТпд}ШуП^ where
(J Ж12 1+72I1	Ж12, 2+^97l |~nd7) < °°-
W-00 log + (2 + |x|) ) U-00 l°g + (2 + |7l) /
If faster decay of g and g is needed, we have to ask whether we need all the properties characterizing a Riesz basis or whether we can relax some of them. The property we want to keep is that every f G L2(K) has an unconditionally convergent expansion in terms of modulated and translated versions of the function g; together with Lemma 3.6.2 this shows that we do not gain anything by asking for {EmTng}mynE% being merely a basis instead of a Riesz basis. However, it turns out that the (unconditionally convergent) expansion property actually can be combined with g and g
4.2 Bases and wavelets
83
Saving very fast decay: the part of the definition of a basis which has to be biven up is the uniqueness of such an expansion. This will bring us from bases to frames. The exact definition will be given in the next chapter, and rthe above description rather tries to state the difference between frames and bases than to give the key to the right definition. Frames having the Gabor structure will be the subject of Chapters 8-9.
. Having a frame of the type {EmbTnag}minez (note that we now allow parameters a, b > 0 in the modulation and translation) one could ask whether there exists a subfamily which is a basis. We will not do so: on page 208 we argue that even if the answer is yes, it will in general not be an advantage to remove elements from {ЕтъТпад}т,пЕ% because the computational benefits from the points {(na, mb)}m>nGz forming a lattice in will be lost.
4.2 Bases and wavelets
Wavelet orthonormal bases {ipj,k}j,k(=z form another important class of bases for L2(K). Also for these bases there are limitations on the properties which can be satisfied simultaneously:
Theorem 4.2.1 Let гр € L2(K). Assume that гр decays exponentially and that	an orthonormal basis. Then гр can not be infinitely often
differentiable with bounded derivatives.
For a proof we refer to [106]. We will see in Example 11.2.7 that the properties in Theorem 4.2.1 can be combined if we allow гр to generate a frame instead of a basis.
For applications the question is not only what is possible mathematically: one also needs to focus on constructions which are convenient to use. As discussed in Section 3.8 this is one of the reasons for constructing wavelet bases via multiresolution analysis. Some of the properties which are relevant for a basis	are
•	that гр has a computationally convenient form, for example that гр is a piecewise polynomial (a spline);
•	regularity of гр;
•	symmetry (or anti-symmetry) of гр, i.e., that гр(х) = гр(—х) or гр(х) = -гр(-х);
•	compact support of гр, or at least fast decay;
•	that гр has vanishing moments, i.e., that for a certain m E N,
/* хЕгр(х)йх = 0 for t = 0,1,..., m.
j —CXO
84
4. Bases and their Limitations
We discuss the role played by these properties and how they motivated the development of wavelet theory below. First, the role of vanishing moments is not immediately clear, but the following proposition shows that a large number of vanishing moments is important if we want to obtain smooth wavelets. For the proof we refer to [106].
Proposition 4.2.2 Assume that 6 L2(JR) is m times continuously differentiable with bounded derivatives, that {^j,k}j,ke% is an orthonormal system, and that there exist constants C, e > 0 such that
IV'WI < (1 + H.W.~ e R.	(4.8)
Then
[ x^'ip^dx = 0 for all I = 0,1,..., m.	(4.9)
J —CXO
The decay condition (4.8) on is automatically satisfied if is a bounded function with compact support, so in this case vanishing moments are unavoidable for to generate a wavelet basis and be differentiable at the same time. This is also the reason that vanishing moments play a role in Daubechies’ construction of compactly supported m times continuously differentiable wavelets. For these wavelets there is a connection between the regularity of the wavelet and the size of the support: the 7V-th Daubechies wavelet has a support with Lebesgue measure equal to 2V - 1, and asymptotically as N -» oo it belongs to C^N with ~ 0.19. We refer to [106] for a proof.
Vanishing moments are also essential in the context of compression. Assuming that	is an orthonormal basis for L2(R), every f € L2(R)
has the representation
j'Aez
All information about f is stored in the coefficients {(/,^j,k)}j,ke%, and (4.10) tells us how to reconstruct f based on knowledge of the coefficients. However, in practice one can not store an infinite sequence of non-zero numbers, so one has to select a finite number of the coefficients to keep. This is usually done by thresholding: one chooses a certain e > 0 and keeps only the coefficients {f^j,k) for which	> e- Here vanishing
moments come in again: one can prove that if has a large number of vanishing moments, then only relatively few coefficients (f^j,k) will be large. By keeping these coefficients and throwing the rest away we have obtained an efficient compression of the signal f. We refer to the paper by Beylkin, Coifman and Rokhlin [27] for more details.
Compact support (or at least fast decay) of is essential for the use of computer-based methods, where a function with unbounded support al
4.2 Bases and wavelets 85
ways has to be truncated. For the same reason we often want the support to be small. The condition of ip being symmetric is less important (or even irrelevant) in many contexts, but there are cases where it is a helpful property; an example is image processing, where a non-symmetric wavelet will generate non-symmetric errors, which are more disturbing to the human eye than symmetric errors. The next result, which is also proved in [106], shows that it is difficult to combine the classical multiresolution analysis with the desire of having a symmetric wavelet ip:
Proposition 4.2.3 Assume that ф 6 L2(K) is real-valued and compactly supported, and let
Vj = span{DJT^}kEZ, j e Z.
Assume that (</>, {V)}) constitute a multiresolution analysis. Then, if the associated wavelet ip in (3.34) is real-valued and compactly supported and has either a symmetry axis or an antisymmetry axis, ip is necessarily the Haar wavelet.
Thus, under the above assumptions we are back at the function we want to avoid! Proposition 4.2.3 was one of the reasons for Cohen, Daubechies and Feauveau to introduce biorthogonal multiresolution analysis [99], where one constructs a Riesz basis {^j,k}j,kez for L2(K) instead of an orthonormal basis. As we have seen in Theorem 3.6.3, the coefficients in the expansion of a function in terms of a Riesz basis are given by inner products between the function and the elements in the dual Riesz basis. This is the reason for the name biorthogonal multiresolution analysis: one actually constructs two multiresolution analyses, which deliver the Riesz basis {^j,k}j,ke% and its dual, which turns out also to have wavelet structure in this special case, i.e., it has the form {ipj,k}j,kez for some function ip 6 L2(K) (this is not obvious; see Section 12.1).
The construction in [99] allows the functions ip, ip to be symmetric and compactly supported, but only one of them can be a spline. A related result by Chui and Wang [98] allows ip as well as ip to be symmetric splines, but only one can have compact support.
This is only a glimpse of the intense activity in the area, which took place around 1990-1993. For our purpose we only mention one more step, which is important for the presentation in Chapters 11-14. This is the idea of using multiwavelets. Here we give up the basic requirement that a wavelet system is generated by translated and scaled versions of one function. In fact, we begin instead with a finite collection of functions ipi,... ,ipn 6 L2(K) and consider the system of functions which we obtain by translation and dilation of all these functions. In [119], Donovan, Geronimo and Hardin proved that one can construct orthonormal bases of multi wavelets, where the functions ipi,... ,ipn are symmetric splines with compact support.
86
4. Bases and their Limitations
From this short description it is clear that the purpose of the different extensions of the first multiresolution scheme is to gain more flexibility. This is also the key reason for extending the theory to frames, as we will do in Chapters 13-14.
4.3 General shortcomings
Another annoying fact about bases is their lack of stability against applications of operators. If, for example,	is an orthonormal basis, then
only very special operators (the unitary ones) will make {Uek}^L} an orthonormal basis. If	is a basis, then we need U to be a bounded
bijective operator in order for {Uek}(^=1 to be a basis. Frames are considerably more stable than bases: application of just a bounded surjective operator will preserve the frame property, as we will see in Proposition 5.3.1. In case we only need expansions of the elements in the range of the operator U, it even suffices that U is bounded and has closed range.
More concrete versions of this statement will also be given. In Example 5.4.6 we prove that if {e^}^ is an orthonormal basis for 7Y, then {ek 4-	does not have the expansion property, despite the fact that
span{efc 4-	= H. The same happens if	is a Riesz ba-
sis (Exercise 5.11). On the other hand, Proposition 7.2.5 will show that {е^ 4-	can very well have the expansion property if	is a
frame.
The limitations on the possible constructions of bases give theoretical reasons to consider frames. We will in Sections 5.9 and 9.8 describe cases where bases actually exist, but where frames simply perform better.
5
Frames in Hilbert Spaces
The main feature of a basis {fk}^! in a Hilbert space H is that every / G H can be represented as an (infinite) linear combination of the elements Д in the basis:
oo / = £q(/)A.	(5.1)
/c=l
The coefficients Ck(f) are unique. We now introduce the concept of frames. A frame is also a sequence of elements {Д}^_1т H, which allows every f G H to be written as in (5.1). However, the corresponding coefficients are not necessarily unique. Thus a frame might not be a basis; arguments for generalizing the basis concept were given in Chapter 4.
The history of frames is a nice example of the development of mathematics. Frames were introduced already in 1952 by Duffin and Schaeffer in their fundamental paper [121]; they used frames as a tool in the study of nonharmonic Fourier series, i.e., sequences of the type {eiXnX}nEz, where {An}nez is a family of real or complex numbers. Apparently, the importance of the concept was not realized by the mathematical community; at least it took almost 30 years before the next treatment appeared in print. In 1980 Young wrote his book [279], which contains the basic facts about frames. Frames were presented in the abstract setting, and again used in the context of nonharmonic Fourier series. Then, in 1985, as the wavelet era began, Daubechies, Grossmann and Meyer [108] observed that frames can be used to find series expansions of functions in L2(JR) which are very similar to the expansions using orthonormal bases. This was probably the
88
5. Frames in Hilbert Spaces
time when many mathematicians started to see the potential of the topic; this point became more clear via Daubechies’ important paper [105], her book [106], and the combined survey/research paper by Heil and Walnut [172]. Since then, the number of papers concerning frames has increased drastically, and a single book can not present all the important results. Our aim is, however, to give an almost complete presentation of all the fundamental results which hold for frames in general Hilbert spaces. The limitations will mainly appear in the later chapters, where we are only able to present some of the many results about Gabor frames and wavelet frames.
A subject like frames can be approached in different ways. One way is to look at frame theory as a branch of functional analysis, and ask what we can prove for general frames in general Hilbert spaces. Another approach is to consider a class of frames, which is used in e.g., signal processing (this could for example be frames having the wavelet structure), and ask what we can prove for this special class of frames. Most papers concentrate on one of these two aspects (the area would actually benefit from a closer coordination) and we will treat them separately here, too. In this chapter we present the general theory, while the next chapters will go into details with specific constructions.
5.1 Frames and their properties
We are now ready to give the central definition.
Definition 5.1.1 A sequence {A}£x of elements in H is a frame for H if there exist constants A, В > 0 such that
oo
л ll/ll2 <El(/,A)|2< в ll/ll2, V/6H.	(5.2)
/c=l
The numbers A, В are called frame bounds. They are not unique. The optimal upper frame bound is the infimum over all upper frame bounds, and the optimal lower frame bound is the supremum over all lower frame bounds. Note that the optimal bounds are actually frame bounds. We collect a few more definitions:
Definition 5.1.2
(i) A frame is tight if we can choose A = В as frame bounds.
(ii) If a frame ceases to be a frame when an arbitrary element is removed, it is called an exact frame.
When we speak about the frame bound for a tight frame, we mean the exact value A which is at the same time an upper and lower frame bound.
5.1 Frames and their properties 89
Note that this is slightly different from the terminology for general frames, where e.g., an upper frame bound just is some number for which the Bessel condition is satisfied.
The definition shows that if {fk}c^=1 is a frame for 7Z, then
span{A}fcli = H.
VJe often need to consider sequences which are not complete in 7Z; they can not form frames for 7Z, but they can very well form frames for the closed linear span of their elements:
Definition 5.1.3 Let	be a sequence in H. We say that {Д}£1
is a frame sequence if it is a frame for 5рап{Д}^Т1.
Before we develop the theory for frames, we mention a few examples of frames. They might appear quite “constructed”, but they are useful for the theoretical understanding of frames. In Chapters 7-14 we consider frames which are more interesting by themselves, for example frames in L2(K) having Gabor structure or wavelet structure.
Example 5.1.4 Let {e/J^ be an orthonormal basis for H.
(i)	By repeating each element in {q}^ twice we obtain
{fkYkLi = {^i, ei, e2, e2,
which is a tight frame with frame bound A = 2. If only ei is repeated we obtain
{A}£i = {^i, e1? e2, ез,..}, which is a frame with bounds A = 1, В = 2.
(ii)	Let
f 1 1 1 1 1 . {fk}k=i {ei, -/=e2, ~2=e2, ~7=e3, -=e3, -^e3, • • • };
that is, {fkis the sequence where each vector is repeated к times. Then, for each f G 7/,
OO	OO	-
Ei</,a>i2 = Efcia,/^i2 k=l	k=l	VK
= ll/ll2-
So {fk}kLi is a tight frame for H with frame bound A = 1.
(iii)	If I C N is a pure subset, then {ek}kei is not complete in 7/, and can not be a frame for 7/. However, {e/J^g/ is a frame for span^/J^g/, i.e., it is a frame sequence.	□
90
5. Frames in Hilbert Spaces
Since a frame {Д}^ is a Bessel sequence, the operator
oo
T : £2(N) H, T{ck}?=1 = '£ckfk	(5.3)
k=l
is bounded by Theorem 3.2.3; T is called the pre-frame operator or the synthesis operator. By Lemma 3.2.1, the adjoint operator is given by
T*:H^£2(N), T*f = {(f,fk)}^.	(5.4)
T* is called the analysis operator. By composing T and T*, we obtain the frame operator
S-.H-+H, Sf = TTrf =	(5.5)
к=1
Note that since {A}^=i is a Bessel sequence, the series defining S converges unconditionally for all f G H by Corollary 3.2.5. We state some of the important properties of S:
Lemma 5.1.5 Let {fk}kLi be a frame with frame operator S and frame bounds A,B. Then the following holds:
(i) S is bounded, invertible, self-adjoint, and positive.
(ii) {S'-1 fk}kLi is a frame with bounds	if A, В are the opti-
mal bounds for {fk}kLu then the bounds B~1,A~1 are optimal for {S-1 fcli- The frame operator for {S"1 fkYjfLt is S-1.
Proof, (i): S is bounded as a composition of two bounded operators. By Theorem 3.2.3,
||S|| = ||JT|I = ||T|| ||r*|| = ||T||2 < B.
Since S* = (ТГ)* = TT* = S, the operator S is self-adjoint. The inequality (5.2) means that A||/||2 < (Sf,f) < B||/||2 for all f G H, or, in the notation from Appendix A.5, Al < S < ВТ, thus S is positive. Furthermore, 0 < Z — B~1S < and consequently
p _ д
11/ - B-1S|| = su^ I<(Z - f)| < -g- < 1,
which by Theorem A.5.3 shows that S is invertible.
(ii): Note that for f G H,
oo	oo
El^s-’A)!2 = £1(5’7, A)l2 < в ||S’7||2
fc=l
< в IIS-1!!2 Il/ll2.
5.1 Frames and their properties 91
T-hat is, {S 1 A}£i is a Bessel sequence. It follows that the frame operator for {S-1 fk}kLi is well defined. By definition, it acts on f 6 H by
oo	oo
= s^ss^f
k=l	fc=l
= $"7;	(5.6)
^his shows that the frame operator for {S-1/fc}£i equals S-1. The operator S-1 commutes with both S and Z, so using Theorem A.6.5 we can ^“multiply the inequality” Al < S < BI with S-1; this gives
B-1Z < S-1 < A~]Z,
i.e.,
B-1 Н/п2 < (5-7,/)< A-1 ll/ll2, v/ew.
Via (5.6),
oo
B-1 ll/ll2	<A-> ll/ll2, VfEH;
k=l
thus {S-1 fk}kLi is a frame with frame bounds B-1,A-1. To prove the optimality of the bounds, (in case A, В are optimal for {fk}fc=i) let A be the optimal lower bound for	and assume that the optimal
upper bound for {5-1Д}£1 is C < By applying what we already proved to the frame {S~} fk}^L} having frame operator S-1, we obtain that {fk}kLi = {(S'-1)“1S“1 fk}^=i has the lower bound > A, but this is a contradiction. Thus {S-1 has the optimal upper bound The argument for the optimal lower bound is similar.	□
The frame	is called the canonical dual of {fk}^i because
it plays the same role in frame theory as the dual of a basis; see Theorem 5.1.6 and Theorem 5.4.1. Frequently we will skip the word “canonical” and just speak about the dual frame; see the discussion on page 112.
The frame decomposition, stated below, is the most important frame result. It shows that if {fk}^=1 is a frame for H, then every element in H has a representation as an infinite linear combination of the frame elements. Thus it is natural to view a frame as some kind of “generalized basis”.
Theorem 5.1.6 Let {fk}^=i a frame with frame operator S. Then
oo
/ = £(/-5’7*)A, V/eH.	(5.7)
fc=l
The series converges unconditionally for all f 6 H.
92	5. Frames in Hilbert Spaces
Proof. Let f G 7Y. Using the properties of the frame operator in Lemma 5.1.5,
oo	oo
f = ss-'f =	= '£^s-1fk)fk.
k=l	k—1
Since {fkYkLi is a Bessel sequence and {(/, 5'-1 A)}£i € ^2(N), the fact that the series converges unconditionally follows from Corollary 3.2.5. □
Similarly, we have the representation
oo
/ = 5-1S/ = £(/,A)S-1A, v/eft-	(5.8)
fc=l
Theorem 5.1.6 shows that all information about the given f e H is contained in the sequence {(/,	• The numbers (f1S~1fk) are
called frame coefficients.
The following lemma shows that it is enough to check the frame condition on a dense set.
Lemma 5.1.7 Suppose that {fk}^LY is a sequence of elements in H and that there exist constants А, В > 0 such that
oo
A ||/||2<El(/,A)|2<5 ll/ll2	(5.9)
fc=l
for all f in a dense subset V of H. Then {fk}^-i is a frame for H with bounds A, B.
Proof. We proved already in Lemma 3.2.6 that {fk}kLi is a Bessel sequence with bound В if (5.9) is satisfied. We now prove that (5.9) implies that the lower frame condition is satisfied on Tl. Expressed in terms of the pre-frame operator T, our assumption means that
A II/Ц2 < IIT’/II2, V/ eV	(5.Ю)
Since T* is bounded and V is dense in it follows that (5.10) holds for all f e H.	□
A note: the proof that the lower frame condition extends from a dense set to H uses the assumption about the upper frame condition being satisfied.
5.2 Frame sequences
Frame sequences and Riesz sequences are useful concepts in cases where we only obtain (or are interested in) expansions in subspaces. As a mathematical example, consider L2(—7Г, 7г) versus L2(IR): restricting the functions
5.3 Frames and operators 93
in a frame for L2(JR) to the interval ] — 7г,7г[ gives a frame for L2(—7г,7г). Qn the other hand, if we extend the functions in a frame for L2(—7Г, тг) to functions in Z/2(IR), by defining them to be zero on JR\] — 7г,тг[, we obtain !a frame sequence in L2(K). Concrete examples of frame sequences appear ^n Chapter 7, where we study frames of translates.
The terminology is also useful in signal processing, where it might be Vnown that the class of relevant signals for a concrete application belongs to a certain subspace of L2(JR) (for example the space of functions whose Fourier transform has support in [—7г,7г]).
We state a criteria for a frame sequence being a frame:
Lemma 5.2.1 Let { A }&S=i be a frame sequence in TL, with pre-frame operator T : £2(N) -» TL. Then {fk}^-^ is a frame for TL if and only if T* is injective.
proof. When {fk}^L1 is a frame sequence we know that
1ZT = span{A}£i-
T* is injective if and only if the range of T is dense in □
If {A}£i is a frame sequence, we can extend Lemma 3.5.2 concerning the Gram matrix:
Proposition 5.2.2 If {fk}kLi is a frame sequence in TL, then the associated Gram matrix defines a bounded invertible operator from the Banach space Ttrr* onto IZt* , with a bounded inverse.
Proof. If { A}&2=i is a frame sequence, then IZt — span{A}£Tp Since TL = IZt Ф = 'R'T ® Ar* , we can write any f G TL as f = T4- z for some {q}^ G ^2(N), z €Thus,
ТЧ = Т*Т{ск}^.
Therefore = T^t* • The rest follows from Lemma 3.5.2 and Theorem A.5.2.	□
Since Ar* — y-
5.3 Frames and operators
Lemma 5.1.5 shows that if {А}£1 is a frame, then {5~1А}£1 is also a frame. This is a special case of a much more general result: {Ufk}^-^ is actually a frame for a large class of operators U. For later reference we state some general versions of this result, where we assume that U is a bounded operator with closed range. We denote the pseudo-inverse (see Lemma A.7.1) of such an operator U by .
94	5. Frames in Hilbert Spaces
Proposition 5.3.1 Let	be a frame for TL with bounds A,B, and
let U : H -» H be a bounded operator with closed range. Then {Ufk}^=1 is a frame sequence with frame bounds A ||[7^||-2, В ||?7||2.
Proof. If f e H, then
oo
E< В IIC7VII2 < В IK ll/ll2, k=l
which proves that {Ufk}kLi is a Bessel sequence. For the lower frame condition, let g 6 span^A}^; we can write g = Uf for some f G span{A}£p By Lemma A.7.2, the operator UlP is the orthogonal projection onto IZu and therefore self-adjoint. Therefore
g = Uf = (UlByUf = (U'yU'Uf.
Thus
к < H(Kll2IPKI2
k=l
fc=l
Thus the lower frame condition is satisfied for all g 6 span{{7A}£i‘> via Lemma 5.1.7 it therefore holds on span{?7fkYjfLi •	О
Exercise 5.7 shows that the conclusion in Proposition 5.3.1 might fail if U does not have closed range. And even if U has closed range, it is not enough to assume that {A}£Li is a frame sequence (Exercise 5.9).
Corollary 5.3.2 Assume that {fk}kL1 is a frame for H with bounds A,B and that U : H H is a bounded surjective operator. Then {Ufk}kLi is a frame for H with frame bounds A 1111“2, В ||t/||2.
In the next result it is enough to assume that {fk}kLi is a frame sequence. We leave the proof to the reader (Exercise 5.10).
Lemma 5.3.3 If {fk}fa=i is a frame sequence with frame bounds A, В and U ’.TL -+TL is a unitary operator, then {UfkjjfL-^ is a frame sequence with frame bounds A,B.
The kind of stability discussed here can often be used to construct frames with special properties. For example it is important to notice that to every frame we can associate a canonical tight frame with frame bound A = 1:
5.3 Frames and operators 95
Theorem 5.3.4 Let {fkjkL} be a frame for H with frame operator S. Denote the positive square root of by S~1^2. Then {S~1^2fk}kL1 is a tight frame with frame bound equal to 1, and
oo
k=l
proof. The existence of a unique positive square root of S-1 follows from Lemma A.6.7. Since S-1/2 is a limit of a sequence of polynomials in S-1, it commutes with S-1 and therefore with S. Therefore every f 6 H can be written
f = S-1/2SS“1/2/ oo
k=l oo
k=l
By taking the inner product with f we obtain that {S~}^2 fk}^L} is a tight (tame with frame bound equal to 1.	□
Orthogonal projections play a special role in many contexts. We state a few relationships between frames and orthogonal projections.
Proposition 5.3.5 Let {A}^ be a sequence in a Hilbert space H, and let P denote the orthogonal projection ofH onto a closed subspace V. Then the following holds:
(i) If {A}fcLi a frame for H with frame bounds A, B, then {Pfk}kLi is a frame for V with frame bounds A, B.
(ii) If{fk}k^i ™ a frame for V with frame operators, then the orthogonal projection ofH onto V is given by
oo
p/ = E(/,5-1A)A, feH.	(5.11)
k=l
The proof of (i) is left to the reader, and the proof of (ii) is identical to the proof of Theorem 1.1.8.
Proposition 5.3.6 Let {fk}kL1 be a frame sequence in H. Then the orthogonal projection Q of a sequence {c^}^ 6 ^2(N) onto the range ofT* is given by
oo	\	00
.	(5.12)
/c=l	/ ) j=l
<?Ь)Г=1 =
96	5. Frames in Hilbert Spaces
Proof. Check via (5.8) that Q defined by (5.12) is the identity on and zero on R?* — Nt-	□
5.4 Frames and bases
Let us now mention some important relationships between frames and Riesf bases. Much more information is contained in Chapter 6.
Theorem 5.4.1 A Riesz basis	for H is a frame for H, and the
Riesz basis bounds coincide with the frame bounds. The dual Riesz basis is {5-74^-
Proof. By Proposition 3.6.4, a Riesz basis {fk}kLi for H is also a frame for if we also involve Proposition 3.6.8, we obtain the statement about the bounds. The rest follows from the frame decomposition combined with the uniqueness part of Theorem 3.6.3.	□
A frame which is not a Riesz basis is said to be overcomplete; in the literature the terms nonexact frame and redundant frame are also used. Theorem 6.1.1 (vii) will explain why the word “overcomplete” is used: in fact, if {fkYj?=i is a frame which is not a Riesz basis, there exist coefficients: {cfc}£°=1 e ^2(N) \ {0} for which
oo
EC*A = °-
k=l
We have already in Theorem 5.1.6 seen that the frame coefficients {(/,	lead to a representation of the given f 6 H. If {Л}^ is an
overcomplete frame, there are other coefficients {qJjEj G ^2(N) for which f — ckfk- As in the finite-dimensional case, the frame coefficients {(/,S-1 fk)}^=1 have minimal ^2-norm among all sequences representing
Lemma 5.4.2 Let	be a frame for H and let f 6 H. If f has a
representation f = Ckfk for some coefficients {ckJ-jEj, then
oo	oo	oo
E led2 - E K/,5-7dl2 + E Iе* - </.-S’"1 A>l2•
A:—1	/c=l	Jc=l
The proof is identical to the proof of Theorem 1.1.5(iii). As a consequence of Lemma 5.4.2 and Theorem A.7.3, we obtain an explicit expression for the pseudo-inverse of the pre-frame operator:
5.4 Frames and bases 97
Theorem 5.4.3 Let {fk}kLi be a frame with pre-frame operator T and frame operator S. Then
The optimal frame bounds can be expressed in terms of the operators T S and their in verses/pseudo-in verses:
proposition 5.4.4 The optimal frame bounds A, В for a frame {fk}^i • are given by
A = ||S-1||_1 = ||Tt||-2, В = ||S|| = ||T||2.
Proof. By definition and using (A.8),
oo
B= sup £|(/,A)|2= sup = ||S||. iifii=i;s	n/n=i
Applying this on the dual frame {S-1 fk}kLi (which has frame operator S-1 and the optimal upper bound by Lemma 5.1.5) we obtain = ||S-1||. For the rest, S = TT* implies that ||5|| = ||TT*|| = ||T||2. Finally, via Theorem 5.4.3 and Lemma 5.1.5,
oo	1
= E KAS-W < -rll/ll2,
k=l
where L js the smallest possible constant; thus ||Tt||2 =	□
In Chapters 7-14 we consider concrete frames for L2(0,1) and L2(R). For the moment we do not go into those constructions, but we just note that already Example 4.0.1 illustrates what is meant by a frame being over complete:
Example 5.4.5 Let us return to Example 4.0.1, where we considered the orthonormal basis {ек}ке% = {e27rlfc:c}fcGz for L2(0,1). Let I C [0,1] be a pure subinterval, |Z| < 1. Since
El(/,efc)|2 = Ц/H2, V/6L2(0,1), /cGZ
the equality in particular holds for all f G L2(Z). So {ек}кЕ% is a tight frame for L2(I). We have already proved that it is overcomplete. However, recall from Lemma 1.6.2 that {ек}ке% is linearly independent.	□
Example 5.4.5 already points to a central property of frames: they can be overcomplete and linearly independent at the same time. The reason for this is the difference between linear independence (i.e., independence of all finite subsets) and cu-independence, as discussed in Section 3.1. For
98
5. Frames in Hilbert Spaces
frames, the correct notion of independence is cj-independence; we return to this point in Section 6.1.
Mislead by the situation in the finite-dimensional setting, one could expect that if span-fZ/c}^ = H, then every f E H would have an expansion f = YlkLi ckfk for certain scalar coefficients	However,
in an infinite-dimensional Hilbert space this does not necessarily hold. We give an example below; the example will be used several times in the sequel.
Example 5.4.6 Let	be an orthonormal basis for H and define
fk efc 4- efc+i, к E N.
Our main purpose is to show that
(i) {A}£i is complete and minimal, and its unique biorthogonal sequence is given by
к	к
gk = 52(-l)n+1en if к is odd, and gk = 52(-l)nen if k is even;
n=l	n=l
(ii) {fk}kLi is a Bessel sequence, but not a frame.
We also derive some consequences of (i) and (ii). First, we prove that {fk}kLi is a Bessel sequence by a direct estimate. For / E H, oo	oo
У?\(f,ek 4- efc+i)|2 = ^2 К/,efc) 4-(/,efc+i)|2	'
fc=l	k=l
oo	oo
< 2£ \(f,ek}\2 + 2£ |</,efc+1>|2 k=l	fc=l
< 4 Ц/H2.	'
To prove that {fk}"}?-! is complete, assume that f E H and that f
{f, ek 4- efc+i) = 0 for all к E N.
Then (/, ek) = -{f, ek+i) for all к E N, implying that |(/, ek)\ is a constant. Since $2/^=1 l(/>e/c)|2 — ll/ll2 < we conclude that (/,ek) = 0, V/c, so f = 0. Thus {fkYkLt is complete. To prove that {fk}£ТХ is minimal, assume the opposite, i.e., that for some j E N,
fj E span{fk}k*j
= span{ei 4- 62,62 4- 63,..., e;_] 4- ej, ej+i 4- ej+2, • • • }
= span{ei 4- 62,62 4- 63,..., ej-i 4- 6j} Ф span{ej_|_fc 4- e;+A;4-i}j*L1.
Since fj = ej 4- ej+1, this would imply that
e3 E span{ei 4- e2,62 4- 63,..., ej-i 4- ej};
5.4 Frames and bases 99
but this certainly does not hold. Thus {fk}™-! is minimal. By Lemma 3.3.1, has a unique biorthogonal sequence {дк}^^ which is determined by the conditions
{дк,ек 4- ek+i) = 1, {дк, e3 4- ej+i) = 0 for j / к. (5.13)
Given к E N, let C := (дк,ек)- Then (5.13) implies that {gk,ek+i) = 1-C, and, in general for j > к, \{gk, ej)| = |1 - C|. Using that {ej}<^=1 is a Bessel Sequence, it follows that C = 1, i.e.,
to,efc) = 1 and = 0 for j > k.
losing again the second condition in (5.13) we are able to find for j < the details are left to the reader. From here, the announced expression for gk follows from gk = ^^{gk^e^ej
To prove that {fk}^^ is not a frame, we apply the frame condition to the elements	in the biorthogonal system. Note that ||pj|| = y/J-
thus
k=l	3
Since this holds for all j E N we see that {fk}kLi does not satisfy the lower frame condition.
The example demonstrates that the union of bases for subspaces might not be a basis or a frame. In fact, {^(e2fc-i 4-	is an orthonormal
basis for span{e2fc-i 4-e2fc}£L1, and {^(e2k + e2k+i)}kL1 is an orthonormal basis for span{e2fc 4- 62^+1 but the union is the family
1	100
~7=(ek + Cfc+i) >	,
W	J k=l
which is not a basis or a frame for
00
It does not even have the expansion property: despite the fact that {fk}^-! is complete, there exists f 6 H which can not be written as f = 52^ ck fk for any choice of the coefficients {с^}^ ! As a concrete example, take f = ei.
There does not exist a basis for H containing {A}£Li as a subset (Exercise 5.4). Furthermore, in Example 15.2.3 we prove that {fk}kL1 can not be extended to a frame for H by adding a finite number of elements. But it follows from (ii) that if a sequence {hk} is a frame, then also {/ifc}U{A}£i is a frame.
Note that an extension is stated in Exercise 5.11: even if {A}fcii is assumed to be a Riesz basis, {A 4- A+i}£ii can n°f be a frame. We will see in Proposition 7.2.5 that the situation is different if {A}£i is allowed to be a frame.	□
span I -^=(ек 4- efc+i)
100	5. Frames in Hilbert Spaces
Theorem 5.4.7 The removal of a vector fj from a frame {fk}^! fori leaves either a frame or an incomplete set. More precisely,
if (faS-'fj) / 1,	then {fk}k& is a frame for'Ll;
if (fjiS~1fj)= 1,	then {fk}k^j is incomplete.
Proof. Choose j G N arbitrarily. By the frame decomposition,
oo
fj =
k—1
Define, for notational convenience, = {fj, S~}fk), so fj =
Clearly, we also have fj = dj,kfk, so Lemma 5.4.2 yields the following relation between 6jtk and ak:
oo
i = Eiw
k=l
Ew2 + f>-<W
k=l
1<ъ12 + У? IM2 + laj ~ 1l2 + У? Ы2.
k^j
We consider the cases aj = 1 and aj 1 separately. First, suppose that aj = 1. Rom the above formula, Y^k±j IM'2 = 0, so that
ak = {S ^jjk} = 0 for all к / j.
Since aj = {S~1fj,fj} = 1 we know that S-1 fj 0. Thus we have found a non-zero element S~rfj which is orthogonal to {fk}k=£j, so {fk}k^j is incomplete.
Suppose now that aj ф 1; then fj —	22^- akfk‘ For any f G H,
Cauchy-Schwarz’ inequality gives
К/, Л>12
5.5 Characterization of frames 101
5vhere С =	1%|2- Let A denote a lower frame bound for
{Ж=1-ТЬеп
oo
a н/п2 < Ek/.m
fc=l
= Ек/,А)12 + К/-Л>12
< (l + C)£|(fM
k^j
showing that {fk}k^j satisfies the lower frame condition with lower bound clearly {fk}fa£j also satisfies the upper frame condition.	□
l+C'
We return to a related result in Corollary 15.2.2. The proof of Theorem 5.4.7 shows an interesting property of {fk}kLi and the dual frame {S'1 fk}kLi^ which will be used later.
Proposition 5.4.8 If {fk}kL1 is an exact frame, then {fk}kLi and {S'1 fk}kLi are biorthogonal and {A}^ is a basis for H.
proof. Assume that {fk}k>=1 is an exact frame and fix j 6 N. Then {fk}k^j is n°t a frame, implying by Theorem 5.4.7 that (fj, S'1 fj) = 1. The proof of Theorem 5.4.7 now shows that (fj^S^fk) = dj,k, i-e., that {fkYf?=1 and {S'1 fk}kLi are biorthogonal. By the frame decomposition we have that every f E H can be expressed as f =	S'1 fk)fk-
In order to show that {fk}^! is a basis, it is enough to show that this representation is unique. But if f = ^^^bkfk for some coefficients bk, then
/ oo	\	oo
(f’S-1 fk) =	= bk. □
\j=l	I	J=1
In Theorem 6.1.1 we will prove more: an exact frame is actually a Riesz basis.
5.5 Characterization of frames
Let us for a moment go back to the definition of a frame. In order to check that a sequence {fk}<^=1 is a frame we have to verify the existence of a positive lower frame bound A and a finite upper frame bound В. Intuitively, the lower frame condition is the most critical to verify: bad upper estimates on l(/> A)|2 will sometimes force us to take a larger value for В than necessary, but bad lower estimates can easily make it impossible to find a
102
5. Frames in Hilbert Spaces
value for A which can be used for all f G H. Later we will see more exact statements, which support this observation: for example, relatively weak decay conditions on a function g 6 L2(IR) will imply that a Gabor system: {EmbTna^}m>nGz satisfies the upper frame condition in L2(K) (Proposition,^ 8.5.2), but no similar statement about the lower frame condition exists, (unless we are allowed to vary the parameter 6, see Proposition 8.5.3).
We now give a characterization of frames in terms of the pre-frame operator, which was proved by Christensen [65]. It does not involve any knowledge of the frame bounds.
Theorem 5.5.1 A sequence {fk}kLi m % a lrarne for % if and only if
oo T	A
к=1
is a well-defined mapping of £2(N) onto H.
Proof. First, suppose that {fk}kLi is a frame. By Theorem 3.2.3, T is a well-defined bounded operator from ^2(N) into H, and by Lemma 5.1.5(i), the frame operator S = 7T* is surjective. Thus T is surjective. For the opposite implication, suppose that T is a well-defined operator from ^2(N) onto H. Then Lemma 3.2.1 shows that T is bounded and that {fk}kLi is a Bessel sequence. Let : H -> ^2(N) denote the pseudo-inverse of T, as defined in Section A.7. For f EH, we have
oo
where (T*denotes the fc-th coordinate of T*f. Thus
ll/ll4 = K/,/)l2
= 1<Е(^/АА,/)|2 k=l oo	oo
<	Ei^w Ek/,a)i2
k=l	k—1
<	IIT+ll2 ll/ll2 fc=l
we conclude that
oo
Ek/,a>i2>^ii/ii2.	□
5.5 Characterization of frames
103
For an arbitrary sequence {fk}*^ in a Hilbert space, spanf/k}^ is itself a Hilbert space, and Theorem 5.5.1 leads to a statement about frame sequences:
Corollary 5.5.2 A sequence {fk}kLi H is a frame sequence if and only if the pre-frame operator is well-defined on ^2(N) and has closed range.
In terms of the adjoint of the pre-frame operator we have:
Corollary 5.5.3 For a sequence {fk}kLi H the following holds:
(i) {fk}kLi is a frame sequence if and only if
(5-14)
is a well-defined map from H onto a closed subspace o/^2(N).
(H) If {fk}kLi is a frame sequence, it is a frame for H if and only if the map (5.14) is injective.
Proof. The proof of (i) uses that a bounded operator has closed range if and only if its adjoint operator has closed range. First, assume that {Л}/Х1 is a frame sequence. Then the pre-frame operator T is well-defined and bounded, and the range is closed. Therefore T* is well-defined, and has closed range. For the opposite implication, if (5.14) maps H into ^2(N), then {fk}&=i is a Bessel sequence (Exercise 3.7). Thus the pre-frame operator T is well defined and bounded; furthermore, if the range of the map in (5.14) is closed, the same is true for T. This implies by Corollary 5.5.2 that {fk}kLi is a frame sequence.
For the proof of (ii) we note that TZt — (Л/т*)1. Thus, if {fkYjfi-i is a frame for H, then T* is injective. On the other hand, if (5.14) defines an injective mapping, then {A}^=1 is complete in H, thus, if {fk}^! is a frame sequence it is a frame for H.	□
Via Theorem 5.5.1 the question of existence of an upper and a lower frame bound is replaced by an investigation of infinite series: we have to check that Ckfk converges for all {с^}^ G ^2(N) and that each f G H can be represented via such an infinite series. The other results mentioned here do not involve the frame bounds either. We now state a characterization of frames which keeps the information about the frame bounds.
Lemma 5.5.4 A sequence {fkjkLi in H is a frame for H with bounds A, В if and only if the following conditions are satisfied:
(i) {fk}kLi is complete in H.
(ii) The pre-frame operator T is well defined on ^2(N) and
oo	oo
а£ы2 < ||T{Q}^=1||2 <вЕЫ2.	(5.15)
k=l	k=l
104	5. Frames in Hilbert Spaces
Proof. Theorem 3.2.3 gives the first part: the upper frame condition with bound В is equivalent to the right-hand inequality in (5.15) (it is clear that it is enough to check the condition for	G A4p). We therefore
assume that {A}^ is a Bessel sequence and prove the equivalence of the lower frame condition and the left-hand inequality in (5.15) together with completeness.
First, assume that {fk}^} satisfies the lower frame condition with bound A. Then (i) is satisfied. Note that is closed because 71т is closed (the latter is equal to H because {fk}^-! is a frame). Therefore
ААр =	,
i.e., A4p consists of all sequences of the form {(AA)}£Li> f G H. Now, given / G
/ °°	\ 2
Ek/,a)i2 = iw,/>i2 \/c=l	/
<	IIS/II2 ll/ll2
1	00
<	iiw 7 Ек/.л>13. .
This implies that
00
A^\(f,fk)\2 < ||S/||2
A:=l
= 11Л(/,Л)}Г=1112-
as desired. For the other implication, assume that {fk}kLi is complete and that the left-hand inequality in (5.15) is satisfied. We first prove that IZt = H. Since span{A}£i C 7&r, it is enough to prove that TZt is closed. Now, if {yn} is a sequence in 72/p, we can find a sequence {тп} in A4p such that yn = Txn\ if yn converges to some у G H, then (5.15) implies that {тп} is a Cauchy sequence. Therefore {xn} converges to some rr, which by continuity of T satisfies Tx = y. Thus IZt is closed and hence 1Zt = Let denote the pseudo-inverse of T. By Lemma A.7.2 and (A.11) we know that the operator T^T is the orthogonal projection onto Af/, and that TT^ is the orthogonal projection onto 71т = H. Thus, for any {q}^ G ^2(N) the inequality (5.15) implies that
A ||TtT{Cfc}^1||2 < ||TTtT{cfc}^=1||2 = ЦТ{с4Г=1 II2-	(5.16)
Again by (A.ll), we have A/74 = 7^^, so (5.16) gives that ЦТ1’||2 < JL. Using Lemma A.7.2, we also have 11 (T* )t 112 < But (T*)1!"* is the orthogonal projection onto
— A/^t — 7Z.T — 'Hy
5.5 Characterization of frames 105
go for all / 6 H, ll/ll(i) 2 = ||(T*)tT*/||2
< 4 нг’/n2
-ТА oo
= 4Ek/,a)i2.
71 k=l
This shows that {fk}(^-1 satisfies the lower frame condition as desired. □
Lemma 5.5.4 is probably most useful for theoretical considerations; see the proof of Theorem 7.2.3 for an application.
Recall that Riesz bases for TL are characterized as the families {Uek}kL1, where	is an orthonormal basis for TL and U : TL -» TL is bounded
and invertible. We can now give a similar characterization of frames:
Theorem 5.5.5 Let	be an arbitrary orthonormal basis for TL. The
frames for TL are precisely the families {Uek}^Llf where U : TL -> TL is a bounded and surjective operator.
Proof. Let	be the canonical basis for ^2(N) and	an
orthonormal basis for TL. Let Ф : TL -» £2(N) be the isometric isomorphism defined by Фе^ = If {A}£1 is a frame, then the pre-frame operator T is bounded and surjective, and T5k = fk. With U := ТФ, we have {fk}kLi = {Uek}%Lv and U is bounded and surjective. That every family of the described type is a frame follows from Theorem 5.5.1 (Exercise 5.14); alternatively, we can observe that
oo
Ek/.^mi2 = \\u*f и2
and refer to Lemma A.6.1.	□
A different approach to determine all frames for TL was given by Aldroubi [1]. Assuming that	is a frame, he considered the questions
(i) Which conditions on the numbers {uik}itke^ will imply that the vectors
oo
Ф1 = ^щк/к, leN	(5.17)
k=l
are well defined and constitute a frame for TL?
(ii) Can all frames for TL be constructed this way?
It is immediately clear that the answer to the second question is yes: the frame decomposition associated with {fk}^! says that for any sequence
106	5. Frames in Hilbert Spaces
{0/}/cE1 in H we can write the elements as Ф1 = 'E^Li'Uikfk with
Ulk =
In case is a frame, the operator defined by	is boundec
Proposition 5.5.6 Let	and	frames for H. Then the
bi-infinite matrix U, where the Ik-th entry is uik = {ф^ S~l fk), defines a bounded operator on ^2(N).
Proof. Let b denote an upper frame bound for and A a lower frame bound for {fk}kLi - The proof will use several frame results. First, we know from Lemma 5.1.5 that {S-1 fk}kLi is a frame with upper bound 1/A. IF follows that {{ф[, S~lfk)}kL1 € f?2(N) for all I € N, and that CkS~lfk is convergent for all {c/c}^_1 € ^2(N). If we also invoke Theorem 3.2.3 we see that for all {q}^ € ^2(N),
№}£il|2 =
£^,5-7^^
□
Concerning the first question Aldroubi proves
Proposition 5.5.7 Let {Д}^_г be a frame and assume that a bi-infinite matrix U = {uik}i,ke^ defines a bounded operator on ^2(N). Then the vec\ tors	in (5.17) are well defined; they constitute a frame for В if and
only if there exists a constant C > 0 such that
00	00
£|Ш/)12 > c £l(AJ)l2, v/6H.
/=1	k=l
(5.18)
Proof. Let В denote an upper frame bound for {fk}k^=i- If U is bounded on ^2(N), then
00
E
/=1
00	“	00
£UlfcCfc < ||t/||2 £ы2, V{ct}£°=1 e m
k=l	k=l
5.5 Characterization of frames
107
{This implies that for any fixed I e N, the map {с^}^=1 -> Sfcli uikCk is a continuous linear functional on ^2(N); since the dual of ^2(N) equals £2(N), we conclude that the rows in the matrix U are square summable, {uikYkLi ^2(N)- Thus the vectors (fa are well defined. By construction,

from here it follows that
El(OI2 = II^{(A-/>}“=1II2
< IK ll{(A,W=1ll2
< IK El(A,/)l2
< В IK ll/ll2-
So {<M£i even a Bessel sequence. Now, if (5.18) is satisfied it is clear that is a frame; on the other hand, if {0/}^ is a frame with lower bound a, then
oo	oo
Ei(0I,/)i2>«ii/ii2>| Eka,/>i2. wm. □
/=1	k=l
The condition (5.18) can also be written as
oo	oo
Е(Л,/Ы >c ElCKI2, vfen.
k=l	k=l
However, it is not clear from here which coefficients {uik}k,ieN will lead to a frame	A sufficient condition on	is given by
Proposition 5.5.8 Let {A}^ be a frame with bounds A,B. If the numbers {uik}k,iew satisfy the two conditions
oo oo
sup EE Wz/cU/J к J = 1 /=1
then	defined by (5.17) is a frame with bounds aA,bB.
108
5. Frames in Hilbert Spaces
Proof. Let f e H. Then
oo
Ei^.ni2
1=1
E (Е«*л,/)
1=1 k=l
OO oo
E Eu^>n
1=1 k=l
oo oo
EEKi2i(A,ni2
1=1 k=l
1=1 k=lj^k
(*) + (**)•
By Cauchy-Schwarz’ inequality we get
!(**)!
oo	oo
EEka,/>(wi E^
k=lj^k	1=1
(oo	oo
EEka>/>i2 E Ulk^lj
k=lj^k	1=1
(oo
EEkwi2
k=lj^k
The two terms in the last product are actually identical. In fact, by switching the order of summation and renaming the indices,
oo
EEkw2
k=l j^k
oo
= ЕЕ к/,ля2
j = l k^j
oo
= ЕЕКЛ-/)!2
k=lj^k
oo
У^ЩкЩ] i=i
oo
^2 Ulkulj
1=1
oo
'U'lk'U'lj
1=1
Thus
oo
(**) < EEka,/>i2
/с = 1 jjLk
OO
'U'lk'U'lj
5.5 Characterization of frames 109
and by the calculation at the beginning of the proof, oo	oo oo
Ei^ni2 > EEki2ka,/)i2
1=1	1=1 k=l
oo	oo
k=lj^k	1=1
oo &«f-£ /=1	j^k
oo > <•£>< fc=l
The upper frame condition is proved similarly.	□
oo
= Ekaj)i2
/c=l
Note that 52^ uikUjk is the inner product between the Z-th and the j-th row in U. This gives a geometric interpretation of the conditions in Proposition 5.5.8: the lower condition e.g., means that the inner product of any row with itself is larger (uniformly over all rows) than the sum of the absolute values of inner products between this row and all other rows.
Casazza proved in [43] that every frame in a complex Hilbert space is a multiple of a sum of three orthonormal bases:
Theorem 5.5.9 Assume that TL is a complex Hilbert space and that {fk}kLi is a for H with pre-frame operator T. Then, for every e E]0,1[ there exist three orthonormal bases	, {e^ }^_1 and {el}^
for H such that
A = ^(^ + e2fc + e3fc), VfceN.
Proof. Let {e/JjE-i denote an orthonormal basis for and let be the canonical orthonormal basis for £2(N). Composing the pre-frame operator for { with the isometric isomorphism from Tl to ^2(N) which maps ek to Jfc, we obtain a bounded linear operator of Tl onto H, which maps ek to fk; by a slight abuse of our standard notation we denote it by T. Given c e]0,1[, consider the operator
1	1 - c T
и.	(5.19)
Since
we see that U is invertible. Using the polar decomposition in Lemma A.6.8 we can write U = VP, where V is unitary and P is a positive operator.
ПО 5. Frames in Hilbert Spaces
Observe that
||P|| = ||V-1C7|| < ||f/|| < ^ + 1-=-^ < 1.
Now we apply the second part of Lemma A.6.8 to write P =	4-
where W,W* are unitary. We can use these decompositions and (5.19) to obtain an expression for the operator T as
r = 777^"^
=
= 1И1 tyw + VW* - I).
It follows from here that
{fk}?=l = {Tek}^=l
= Ж (VW{ek}f=l + VW*{ek}^ - {ek}?=l).
Since VW and VW* are unitary, the result now follows from Theorem 3.4.7.	Г
Note that on the other hand we can not conclude that every sum of three orthonormal bases is a frame. Consider for example the one-dimensional Hilbert space C; each of the complex numbers 1, e-^, e~^ constitutes an orthonormal basis for C, but 1 4- e-^ 4- e“^ =0.
In case H is a real Hilbert space, the representation of a positive operator P with |\P\| < 1 as an average of two unitary operators is no longer valid. However, a representation as a sum of 16 unitary operators holds. Inserting this in the above proof we obtain a representation of a frame as a sum of 17 orthonormal bases!
Theorem 5.5.9 is optimal in the sense that two orthonormal bases are not enough to represent all frames:
Example 5.5.10 Let	be an orthonormal basis for P, (a real or
complex Hilbert space) and define the frame {fk}kLx by
fi = 0, fk = ek-i, к >2.
Assume that we could find orthonormal systems {e^}^=1,{e|}^_1 and nonzero constants a, b such that fk = aej, 4- be? for all k. Then in particular /1=0 = ae]+bel, which implies that span ej = span ej. By orthonormality we conclude that
span e} C span ({e£}j*2 U {e£}£l2)± •
5.6 The dual frames
111
Qn the other hand
H = span{/*}gl2 = span{aej. +
which is a contradiction.	□
Using more advanced tools from operator theory one can prove that the class of frames which can be written as a linear combination of two orthonormal bases is exactly the class of Riesz bases. We refer to [43] for the proof.
5.6 The dual frames
In Lemma 5.4.2 we have seen that the frame coefficients have minimal (?-norm among all sequences, which represent an element f E H in terms of a given frame {fk}kLi- However, minimality of the ^2-norm of the coefficients {q}^i in the expansion
oo f = 52Ck^ k=l
is not always an important issue; there are cases where other criteria are more relevant. For this reason, it is natural to exploit the freedom in the choice of the coefficients {с^}^ and search for alternatives to the traditionally used frame coefficients {(/,
Usually we want to work with coefficients which depend continuously and linearly on f; by Riesz’ representation Theorem A.6.3, this implies that the /с-th coefficient in the expansion of f should have the form ck(f) = if, 9k) for some gk E TL. If	is an overcomplete frame, there always exist
several choices for {дкУ^'
Lemma 5.6.1 Assume that {fk}kLi is an overcomplete frame. Then there exist frames {дк}^=1 #	for which
oo
/ =	V/eH.	(5.20)
k=l
Proof. We split the proof in two cases, and assume first that fa = 0 for some V, E N; in this case = 0. By letting gk := S~rfk for к I and choosing gt to be an arbitrary non-zero vector, the frame decomposition shows that (5.20) holds, and {gk}kLi £
Now we consider the case where fk / 0 for all к E N. We will use a result proved later, namely in Theorem 6.1.1: it says that since {fk}kLi is overcomplete, there exists a sequence {q}^_1 E f?2(N) \ {0} such that
oo
° = 52
k=l
112	5. Frames in Hilbert Spaces
For a certain e N we have c£ 0, and we can write
Thus {fk}k^e is complete in H, and therefore a frame by Theorem 5.4.7. Denoting its canonical dual frame by {дк}к& and defining g£ = 0, we have found a frame	for which (5.20) holds; it is different from the
canonical dual of {fk}kLi because S~} ft 0.	□
A frame {gk}^=i satisfying (5.20) is called a dual frame of {fk}kLi- To avoid later confusion, we note that most of the results in this book concern the canonical dual', in cases where no confusion can arise we will just call it the dual, despite the fact that other duals might exist.
Following Li [207] we now aim at a characterization of all dual frames {9k}^i associated to a given frame {fk}^- Since {fkYkLi and are assumed to be Bessel sequences we can consider the pre-frame operators; as usual we denote the pre-frame operator for {fk}kLl by T, and we denote the pre-frame operator for {pfc}^=1 by U. In terms of these operators (5.20) means that
TU* = I.
We begin with a lemma, which shows that the roles of {fk}kLi and can be interchanged and that the lower frame condition automatically is satisfied for Bessel sequences {A}^5=1, {gkYk=i if (5.20) holds.
Lemma 5.6.2 Assume that	and	are Bessel sequences in
TL. Then the following are equivalent:
(i)	Wen.
(ii)	We-H.
(m) (f,g) =	W,ge n.
In case the equivalent conditions are satisfied,	and	are ,
dual frames for TL.
Proof. In terms of the pre-frame operators (i) means that TU* = I; this is equivalent to
UT* = I,	(5.21)
which is identical to the statement in (ii). It is also clear that (ii) implies (iii). To prove that (iii) implies (ii) we fix f G TL and note that
is well defined as an element in TL because {A}£2-! and
5.6 The dual frames 113
Bessel sequences. Now the assumption in (iii) shows that /	00	\
\	fc=l	/
and (ii) follows.
In case the equivalent conditions are satisfied, we can write
00
ll/ll2-(/,/) = Е</,<^(/ь/), v/ен.
"Using Cauchy-Schwarz’ inequality and that one of the families {fk}kLi, {ghYkLi is a Bessel sequence, we obtain that the other family satisfies the lower frame condition.	□
When (5.21) is satisfied, we say that U is a left-inverse of T*.
Lemma 5.6.3 Let {fkYjfLi be a frame for H and	be the canonical
orthonormal basis for ^2(N). The dual frames for {fk}kLi are precisely the families {gk}^=1 = {V5k}^Ll} where V : ^2(N) -» H is a bounded left-inverse of T*.
Proof. If V is a bounded left-inverse of T*, then V is surjective; by Theorem 5.5.1 it follows that	:= {V5fc}^.x is a frame. Note that
in terms of
00 т‘/ = {(/,А)}Г=1 = Е</,а^; fc=l
thus, for all f E H,
00 f = VT*f = ^{f,fk)gk, k=l
i.e., {gk}^! is a dual of {Д}^_1. For the other implication, assume that {dk}kLi is a dual frame of {fk}^=i- Then the pre-frame operator U for {dk}kLi satisfies the conditions: in fact, {gk}kLi = {U6k}<^-1, and by Lemma 5.6.2, UT* =1.	□
Lemma 5.6.4 Let {A}jE-i be a frame with pre-frame operator T. The bounded left-inverses of T* are precisely the operators having the form S~lT + W(I — T*S~1T), where W : ^2(N) —> TL is a bounded operator, and I denotes the identity operator on ^2(N).
Proof. Straightforward calculation gives that an operator of the given form is a left-inverse of T*. On the other hand, if U is a given left-inverse of T*, then by taking W = U,
S~YT + W(I - T'S-LT) = S-'T +U - UT*S~1T =U.	□
114
5. Frames in Hilbert Spaces
We are now ready for the announced characterization of all dual frame associated to a given frame.
Theorem 5.6.5 Let {fk}kLi be a frame for LL. The dual frames of are precisely the families
°° 00
«=1 = 1 S-lfk + hk-	,	(5.22)
where	is a Bessel sequence in Tt.
Proof. By Lemma 5.6.3 and Lemma 5.6.4 we can characterize the dual frames as all families of the form
= {S-'TSb + W(I - Т*5-1Т)<5*}^.1>	(5.23)
where W : ^2(N) -> LL is a bounded operator, or, equivalently, an operator of the form	СЛ where {Нк}&-1 is a Bessel sequence.
By inserting this expression for W in (5.23) we get
{9k}Zi = {S~lfk + W5k- WT'S-'T^}^
OO
00	I
S-'fb + hb-^S-'hJjjhA .
j=1	J k=l
□
Note that if {A}^L1 is a Riesz basis, then {fk}kLi and {S'-1 fkYjfLi are biorthogonal by Theorem 5.4.1. Thus, independently of the choice of we have (S'1 Д, fj)hj = hk', that is, Theorem 5.6.5 gives that the unique dual is {S-1 which is in accordance with Theorem 3.6.3.
Given a frame	one could also ask for a characterization of all
families {gk}kLi (frames or not) such that (5.20) holds. We shall not go into this subject, but ask the reader to think about the different possibilities (Exercise 5.18). We also mention the paper [209], where a non-frame {9k}kLi satisfying (5.20) is found for a frame {fk}^=i in the context of wavelets.
In Chapters 11-14 we will see an important reason for considering other duals than the canonical. In fact, we will study frames which have a convenient special structure (namely, wavelet structure), and unfortunately it turns out that the canonical duals might not have the same structure. However, often one can find other duals having the same structure as the frame itself. See in particular Section 12.1 and Section 14.8.
5.7 Tight frames 115
5.7	Tight frames
Tight frames are very attractive from the computational aspect, as well in the finite-dimensional case as in the infinite-dimensional case. The cumbersome inversion of the frame operator, which for general frames is needed e.g., if we want to use the frame decomposition, is avoided: the canonical dual of a tight frame {fk}^ with frame bound A is simply
This implies many other advantages. For the design of frames with prescribed properties it is essential to control the behavior of the dual frame, but the complicated structure of the frame operator and its inverse makes this difficult. If, for example, we consider a frame for L2(R) consisting of functions with exponential decay, nothing guarantees that the functions in the dual frame have exponential decay. However, for tight frames questions of this type trivially have satisfying answers. Also, for a tight frame, the dual automatically has the same structure as the frame itself; what this exactly means will become clear in Section 12.1.
For later use we state an observation relating tight frames to the question of finding dual frames.
Lemma 5.7.1 Let {A}^T1 be a frame. Then the following are equivalent:
(i)	{fk}^ is tight.
(H) {fk}kLi has a dual of the form дь = Cfk for some constant C > 0.
Proof. (i)=>(ii) follows by letting {дь}^} be the canonical dual of {fk}kLi'i (ii)=^(i) follows from Lemma 5.6.2 by taking f = g.	□
5.8	Continuous frames
A generalization of frames was proposed by Kaiser [197] and independently by Ali, Antoine and Gazeau [5]:
Definition 5.8.1 Let H be a complex Hilbert space and M a measure space with a positive measure p. A continuous frame is a family of vectors {fk}keM for which
(i)	for all f EH, к —> (/, fk) is a measurable function on M;
(ii)	there exist constants A, В > 0 such that
Л ll/ll2 < f \{fJk)\2dp(k)<B\\f\\\ VfeH. J м
Note that Kaiser used the terminology generalized frames. Also, since {fk}keM being a continuous frame or not depends on the measure space,
116	5. Frames in Hilbert Spaces
one should be more careful and speak about a continuous frame for H with respect to the measure space
The frames considered so far correspond to the case where M = N, equipped with the counting measure. An important feature of continuous frames is that the frame theory considered so far and some results on the continuous Gabor transformation and the wavelet transform can be considered as different manifestations of a single theory. We come back to this in Chapters 8 and 11.
Let us derive the basic results for a continuous frame {fk}keM- First, Cauchy-Schwarz’ inequality shows that the integral fM(f, fk) (fk, 9)dp(k) is well defined for all f,g E H. For a fixed f E H, the mapping
9^ [ (fJk){fk,9)dp(k)
J м
is clearly conjugated linear, and bounded because
[ {f,fk){fk,g)dti(k)2 < [ |(/,A)|2<W) [ |(А,<<Ш)
J M	J M	J Л/
< в2 H/ll2 1Ы12.	(5.24)
By Riesz’ representation Theorem A.6.3 there exists a unique element in H - we call it fM(f, fk)fkdp(k) - such that
([ (ШМ^к),д\= [ {f,fk)(fk,g)dn(k')
\Jm	I Jm
for all g E H. By this procedure we have defined a mapping	1
Sf=[ (fjk)fkdp(k).
J м
It is easy to check that S is linear; using that
||S/||= sup \(Sf,g)\
it follows by (5.24) that S is bounded and that ||S|| < B. By definition, S is positive,
A H/Ц2 <	< В Ц/H2, V/GB;
exactly as in the proof of Lemma 5.1.5 one can now prove that S is invertible. Thus, every f E H has the representations
f = S~1Sf=l'(f,fk)S~1fkdp,(k),
Jm
f = SS~1f= [ l^S-'f^hd^k).
J M
Remember that these representations have to be interpreted in the weak sense. Sometimes stronger results (like pointwise convergence if 7Y = L2(IR)) can be obtained in concrete cases.
5.9 Frames and signal processing 117
Continuous frames will only appear in Section 8.1, Section 11.1, and Section 17.1.
5.9	Frames and signal processing
The frame theory described so far takes place in an ideal world, which can hardly be realized in e.g., signal processing. In this short section we describe some of the steps which have to be taken in order to apply the abstract results in practice. Much more can of course be said about this important subject, and we refer to the books [218] by Mallat, [265] by Strang and Nguyen, and [282] by Vetterli and Kovacevic for more detailed information.
Some of the problems appear before one even thinks about frames. In fact, even the most basic ingredient in mathematics, the real numbers, are disturbed when applying computers: every number has to be replaced by a number with finitely many digits before processing. This means in practice that we represent all numbers in an interval (for example [1,1 4- 10-18[) by the same number (in this case the number 1). The consequence is an inaccuracy, which is called the quantization error.
The basic limitation in applications of the frame results is that any type of signal processing has to be performed on finite sequences of numbers. This implies for example that the frame representation (5.1.6) has to be truncated: we can only aim at calculating a finite number of frame coefficients, say, {(/, S-1	and the exact representation in (5.1.6) has to
be replaced by
N
k=l
Even calculation of the frame coefficients (/, S~rfk) can in general only be done with finite precision. That is, the outcome of a calculation will be
(f'S-'fkj+wk	(5.25)
for some (hopefully small) error term All types of transmission or further processing will introduce extra inaccuracies. One says that the frame coefficients (/,5-1Д) have been contaminated by the noise
We have already on page 7 given a rather intuitive argument that overcompleteness of frames might reduce the influence of noise, compared to use of an orthonormal basis. To support this further we shortly discuss a result which is proved in [218].
Let us again use the example of signal transmission, as on page 7. That is, we assume that one wants to transmit the signal f from A to 7Z by sending the frame coefficients {(/, S~l fk)}kLi • Due to quantization, the coefficients will be contaminated by some noise	, and 1Z will receive
118	5. Frames in Hilbert Spaces
+ wfc}bi! we assume that {w}^ E ^2(N). The receiver TZ will believe that the transmitted function was
oo	oo
E	+wfc) fk = f + ^wkfk
k=l	k=l
rather than f.
Note that 7Z actually knows that the transmitted sequence was supposed to be a sequence of frame coefficients, i.e., a sequence of the form {(#, A)}a5=i f°r some g E H (namely, g = S~lf). This might not be the case with the coefficients in (5.25), so it is natural to compensate for this by projecting the sequence {(/,5-1Д) +	onto the range of the
operator T* by the operator Q in Proposition 5.3.6. The result is
Q{{f, S-1 fk) + wfc}-, = {{f^fkY^+Qw
with w = {wfc}^. Based on this modification, 7Z will reconstruct the transmitted signal as
oo	oo
E «/, S-1 fk} + (Qw)k) fk - f + £((?w)fc fk.
k=l	k=l
Assuming that the quantization error is white noise (which is realistic), its energy is measured by the mean value of the random variable |w|2, which is denoted by E|w|2. Now, it is proved in [218] that increased redundancy of the frame, measured by a larger lower frame bound, will decrease the energy of the coefficients in the noise:
Proposition 5.9.1 Suppose that the frame	consists of normalized
vectors. Consider the error term in (5.25) as white noise, and assume that w —	has zero mean, Ew = 0. Then
E|Qw|2 < 1e|w|2. /1
The relevance of the result is that it might be easier to increase the redundancy of the frame than to increase the quantization precision.
Quantization errors and noise during transmission are just some of the obstacles for frames in real life. Depending on the underlying Hilbert space Tl there might be additional complications. In many cases H will be a function space like L2(R), and even finite-dimensional subspaces hereof can not be processed directly: a discretization step is needed in order to transfer the setting to a finite-dimensional sequence space like C71. This is exactly the point where the importance of the Gram matrix becomes clear: while the frame operator S = TT* maps H onto itself, the Gram matrix T*T is an operator on the sequence space £2(N), i.e., the only step which is needed is a truncation. For this reason it is an advantage to formulate algorithms involving frames in terms of the Gram matrix rather than the frame operator, if possible.
5.10 Exercises 119
5.10 Exercises
5.1	Prove that the upper and lower frame conditions are unrelated: in an arbitrary Hilbert space H there exists a sequence {fk}^^ satisfying the upper condition for all f E H, but not the lower condition; and vice versa.
5.2	Let	be an orthonormal basis and consider the family
{fkYkLi {ei 4- ^ek, 6k}£L2-
(i)	Prove that {fk}^-1 is not a Bessel sequence.
(ii)	Find all possible representations of e± as linear combinations of
(iii)	Prove that there exists no set of coefficients having minimal ^-norm among all sequences representing e±.
5.3	Give an example of a frame	for which 52^ Ckfk converges
for some	£ ^2(N) (compare with Exercise 3.9!).
5.4	Show that the family {efc 4- е^+1}^=1 in Example 5.4.6 can not be extended to a basis for H.
5.5	Let H be the complexification of a real Hilbert space H- Prove that a frame for H also is a frame for H.
5.6	Let {fk}kLi be a Riesz basis with bounds Prove that
A < ЦАЦ < В for all к E N,
and that the elements in the dual Riesz basis {gk}^-! satisfy
4 < ||р*|| < -T for all k e N.
£>	A
5.7	Prove that the conclusion in Proposition 5.3.1 might fail if U is not assumed to have closed range. (Hint: let	be an
orthonormal basis and define U by Uek = ek 4- e^+i.)
5.8	Find an example of a sequence in a Hilbert space which is a basis but not a frame.
5.9	Let {efc}^=1 be an orthonormal basis for and define an operator U on H by
Ue2k = Le2fc+i = 762^, к E N.
к
Prove that
120
5. Frames in Hilbert Spaces
(i)	U is a well-defined bounded operator on %, and 1Zu is closed.
(ii)	{e2fc+i}fcLi is a frame sequence, but {Ue2k+i}kLi is not.
Thus Proposition 5.3.1 does not extend to frame sequences.
5.10	Prove Lemma 5.3.3.
5.11	Assume that {fk}kL± a Riesz basis. Prove that {fk + А+1}£Т1 can not be a frame.
5.12	Let {fk}ke% be a Riesz basis. Our purpose is to show that
{fk 4- /fc+i}fcez
can not be a frame; compare with Exercise 5.11. Let {gk}kez be the biorthogonal basis associated with {fk}kez- Let
(i)	Prove that	fk + A+i)|2 = 2.
(ii)	Prove that ||/ij||2 > j/В, where В is an upper frame bound for {fk}kez-
(iii)	Conclude that {fk 4- A+iHez is not a frame.
5.13	Prove that the conditions in (A.ll) are equivalent to the construction of the pseudo-inverse in Lemma A.7.1.
5.14	Prove via Theorem 5.5.1 that {Uek}%Li is a frame whenever {efe}fc^i is an orthonormal basis and U is a bounded surjective operator.
5.15	Prove Proposition 5.3.5.
5.16	Let {fk}kLi =	be a Riesz basis for H as in Definition
3.6.1. Prove that the frame operator for {fkYj£=i is given by
S = UU*.
5.17 Let {fk}kLi be a frame with frame bounds A,B. Show that the frame operator S satisfies the inequalities
A Ц/Il < IIS/Ц < В H/Ц, V/ G U.
5.10 Exercises 121
5.18 This exercise concerns the question of finding generalized duals which are not frames.
(i) Find an overcomplete frame {Д}^, for which a family {gkj^-i satisfying (5.20) automatically is a frame.
(ii) Find an overcomplete frame	and a non-Bessel sequence
{9к}^=1 such that (5.20) is satisfied.
6
Frames versus Riesz Bases
As we have seen, a frame {Л}^=1 in a Hilbert space has one of the main properties of a basis: given / € Й, there exist coefficients	€ /?2(N)
such that f = Y^kLickfk- This makes it natural to study the relationship between frames and bases. We have already seen that Riesz bases are frames. In this chapter we exploit the relationship between these two concepts further. In particular, we give equivalent conditions for a frame to be a Riesz basis.
r Intuitively, we think about a frame as some kind of “overcomplete basis”. It turns out that, in the technical sense, one has to be careful with such statements. In fact, we prove the existence of a frame for which no subfamily is a frame. On the other hand, sufficient conditions for a frame to contain a Riesz basis as a subfamily are also given.
6.1 Conditions for a frame being a Riesz basis
Recall from Lemma 3.1.3 that oj-independence and minimality are two different concepts. We now give some equivalent conditions for a frame to be a Riesz basis; in particular, we prove that for a frame oj-independence is equivalent to minimality.
124
6. Frames versus Riesz Bases
Theorem 6.1.1 Let {fk}(^L1 be a frame for Then the following are equivalent.
(i) {A}£i 25 a Riesz basis for H.
(ii) {fk}kLi is an exact frame.
(Hi) {fk}k>-i is minimal.
(iv)	{fk}kLi has a biorthogonal sequence.
(v)	{fk}kLi and {S-1 fk}^=i are biorthogonal.
(vi)	{fk}kLi is ш-independent.
(vii)	If 52^! Ckfk = 0 for some {c/J^ € ^(N), then c* = 0, Vfc € N.
(viii)	{fk}^i ™ a basis.
Proof.
(i)=>(ii) A Riesz basis {fk}kLi is an exact frame: if an arbitrary element is removed, the remaining family is not complete, and therefore not a frame.
(ii)=>(v). This is Proposition 5.4.8.
(v) => (iv) is clear, and (iv)=> (iii) is proved in Lemma 3.3.1 (i).
(iii)=>(ii). Assume that {A}^ is minimal. Then, for an arbitrary j e N, the family {fk}k^j is incomplete in H, and therefore not a frame for H.
(ii)=>(i). If {fkYkLt is an exact frame, it is a basis by Proposition 5.4.8. Let as usual	be the canonical basis for £2(N). The pre-frame operator T : ^2(N) —> H associated with	is bounded and surjective by
Theorem 5.5.1, and Tdk = fk','m order to show that {fk}fc=i is a Riesz basis it is enough to prove that T is invertible, and this follows from {fk}k^=( being a basis.
(i)=3>(vi). Assume that {fk}kLi is a Riesz basis and that ^2^ Ckfk = 0 for a given sequence {q}^ of scalars. Then by the result in Exercise 3.9, {cfc}^-! € £2(N). Denoting a lower Riesz bound by A, Exercise 3.9 also shows that
oo
-4£kl2<
A:=l
oo
^Ckfk
fc=l
thus Ck = 0 for all к.
(vi)=>(vii). Clear.
(vii)=>(i). Let again	be the canonical orthonormal basis for
£2(N). The assumption (vii) assures that the pre-frame operator T is injective, and T is also surjective because {fkjkLy is a frame. Since Tdk = fk, Чк, the result follows from the definition of a Riesz basis.
(i)=>(viii)=>(iv). Clear.	□
6.1 Conditions for a frame being a Riesz basis 125
Note the slight difference between the conditions (vi) and (vii) in Theorem 6.1.1. As seen in Exercise 5.3, one difference between frames and Riesz bases is that for a Riesz basis	the series 52^ Ckfk is only
convergent for {q}^ E ^2(N); for general frames the series might converge for coefficients {q}^ £ £2(N). However, to check whether a frame ,.is cj-independent, it is enough to consider ^-sequences.
Lu-independence is a stronger condition than just linear independence. To t illustrate that point, we prove one more characterization of Riesz bases. The result should be compared to the equivalence of (i) and (vi) in Theorem 6.1.1- The equivalence (i) <=> (ii) below first appeared in [199], and the equivalence (ii) <=> (iii) is from [81].
Proposition 6.1.2 Let {A}^ be a frame for H. For n E N, let An denote the optimal lower frame bound for the frame sequence {fk}k=i- Then the following are equivalent:
(i) {fh}kLi "is a Riesz basis for H.
(ii) {fk}kLi linearly independent and infnGM An > 0.
(Hi) {fkjkLi is linearly independent and limn_>00 An exists and is positive.
Proof. For the proof of (i)=>(ii) we note that any basis is linearly independent. Assume that {A}^=1 is a Riesz basis, with lower Riesz bound A. Then each subfamily {Д}^_1 is a Riesz sequence, and An is also the optimal lower Riesz bound by Theorem 5.4.1; thus A < An for all n E N and (ii) follows. For the proof of (ii) => (i), assume that (ii) is satisfied. Then, for each n E N, {fk}k=i a (Riesz) basis for its span with lower frame bound A := infnG^ An. Letting В denote an upper frame bound for {A}£i> Theorem 3.2.3 shows that
л£ы2<
/с=1
n
^>2 Ckfk
k—1
<вЕы2
k=l
(6.1)
for all scalar sequences	By Theorem 3.6.6 we conclude that
{AJbLi is a Riesz basis for H. That (ii) <=> (iii) follows from the fact that when {fk}^Li is linearly independent, the sequence of optimal frame bounds An, n E N, is equal to the sequence of optimal lower (Riesz) basis bounds, which is decreasing by definition.	□
Note that we already in Example 5.4.5 saw a natural example of an overcomplete frame which is linearly independent.
126
6. Frames versus Riesz Bases
6.2 Riesz frames and near-Riesz bases
Several variations on the definition of frames and Riesz bases are possible
Definition 6.2.1 Let {/fcjjg-i be a sequence in H. We say that a frame {/*}& ™
(i) a near-Riesz basis if it consists of a Riesz basis and a finite number of extra elements;
(ii) a Riesz frame if every subfamily of {fk}(^L1 is a frame sequence, with uniform frame bounds A, В.
Near-Riesz bases are relevant because they share many properties with Riesz bases. For an arbitrary sequence {fkYkLi in %, the excess is defined as
е({А}£1) := sup{|J| : J C N and span{fk}k&i\j = span{fk}T=i}-
For a near-Riesz basis, the excess is equal to the number of elements which have to be removed in order to obtain a Riesz basis. Holub proved in [178] that for a near-Riesz basis {fk}kLi with pre-frame operator T,
е({А}Г=1)=а1тЛ?.
To motivate the definition of Riesz frames, we recall Corollary 3.6.7: a subfamily of a Riesz sequence is a Riesz sequence. For frames the situation is different. Consider for example the frame in Example 5.1.4 (ii); it contains the subfamily {^ek}(^-1, which is not a frame sequence. Riesz frames were introduced in [70] in order to avoid this situation. In the sequel we will see that Riesz frames behave like Riesz bases in many contexts, despite the fact that a Riesz frame can be overcomplete.
Note that the concepts of Riesz frames and near-Riesz bases are unrelated (Exercise 6.1 and Exercise 16.1).
6.3 Frames containing a Riesz basis
Intuitively, we think about a frame as some kind of “overcomplete basis”, so a natural question is the following: given a frame {fk}^i, is it possible to extract a basis {fk}keJ, J C N, from {fkYkLi, i.e., does {fk}^^ contain a basis as a subset?
Clearly, the answer depends on which kind of basis we are interested in. In this section, we shall find sufficient conditions for a frame to contain a Riesz basis. In the next section we construct a frame consisting of vectors having norm one, which does not even contain a Schauder basis.
6.3 Frames containing a Riesz basis
127
Lemma 6.3.1 Let {q}^ be a sequence of nonnegative numbers for which ck < 00 Suppose that {JnjneN is a family of subsets ofN such that
(i) Л 3 J2 2 <73 D...
(ii) There exists c > 0 such that c < ^kEJn ckl Vn 6 N.
Then c < Skenj„cfc •
proof. Define a positive measure p on the сг-algebra of subsets of N by
m(5) = J>.
kes
By Lemma A.3.3,
м(Л) = Cfc -+ д(гип) = E Cfc as n -> oo, keJn	kenjn
;and the result follows.	□
Our purpose is to show that every Riesz frame contains a Riesz basis, a result which appeared in [70]. Our proof is based on the axiom of choice, also known as Zorn’s Lemma :
Lemma 6.3.2 Let M. be an ordered set. If every totally ordered subset of M has an upper bound in M., then Л4 has at least one maximal element.
Theorem 6.3.3 Every Riesz frame contains a Riesz basis.
Proof. For convenience we use the index set N. Let {A}^ be a Riesz frame, and let A be a common lower bound for all its subframes. Consider the set
A't-HAheJ I JCN, and A||/||2<52K/,A>|2, V/ E H 1. (6.2)
I	keJ	J
Л4 is non-empty. Define an order on Л4:
{fk}ktK К C J.
Now consider a totally ordered family of elements { fk}ke Jn E Л4, n in some index set I. Such a family has an upper bound {fk}kenjn, which is still an element from Л4; to see this we have to prove that
a ll/ll2 < E l(/,A)l2, v/eH.	(6.3)
fcenjn
In case the index set I is countable, (6.3) follows from Lemma 6.3.1; for an uncountable index set the result is still true, but a more general measure-theoretic argument is needed (we skip the argument; see e.g. [225] for
128
6. Frames versus Riesz Bases
arguments of this type). So, by Zorn’s Lemma 6.3.2, Л4 has a maximal element {fk}ke J- Now we show that {A}fcej is a Riesz basis. Clearly {fk}k^j is a frame, so by Theorem 6.1.1 it is enough to show that {fk}k^J is independent. But if not, we could find an element /n,n E J such that {fk}keJ-{n} was still complete, and therefore a frame for H by Theorem 5.4.7. That is, {fk}keJ-{n} £ Л4, which contradicts the maximality of {fk}kej-	□
Via iterated application of Theorem 6.3.3 one can show that every Riesz frame is the union of a finite number of Riesz sequences. We refer to [83] for details.
The result in Theorem 6.3.3 can be slightly generalized: using a more complicated argument, it is shown in [51] that the conclusion holds for any frame {A}^=1 with the property that each subfamily {fk}keJ, <7 C N is a frame sequence. Such a frame is said to have the subframe property.
Lemma 6.3.4 Assume that a frame {fk}kLi forH has the subframe property. Then a subfamily {fk}keJ is a Schauder basis for H if and only {fk}keJ is a Riesz basis for H.
Proof. Assume that {fk}keJ is a Schauder basis for some J CN. By the subframe property, {fk}keJ is also a frame, so we conclude by Theorem 6.1.1 that {fk}kej is a Riesz basis.	□
Lemma 6.3.4 does not hold if the assumption of {A}£i having the subframe property is removed (Exercise 6.2).
6.4 A frame which does not contain a basis
Even if {fk}kLi is not a Riesz frame, the proof of Theorem 6.3.3 shows that the set Л4 in (6.2) will contain a maximal element for every lower frame bound A. However, this element need not constitute a Riesz basis. For example, let	be an orthonormal basis and consider the frame
from Example 5.1.4,
{/d£i := {ei, -U2, -U2, ^e3, -U3, -U3) }.	(6.4)
v Z	v z v <5 v <5 у u
No matter how small A is chosen, the set Л4 does not contain a Riesz basis
for this frame. In fact, for any e > 0 and any {fk}keJ € Л4, the condition
e ll/ll2 <El(/,A)|2, v/€-h
k^J
implies that -^en appears more than once in {fk}k<=j for large values of n.
Thus Ad does not contain an cj-independent subset. Actually	does
6.4 A frame which does not contain a basis 129
not contain a Riesz basis at all. The only candidate would be {^=ek}^11 which is a Schauder basis but not a Riesz basis.
Now we want to show that it even is possible to construct a frame which consists of vectors that are norm bounded below, but which does not contain a Schauder basis; in particular, it does not contain a Riesz basis. The example was constructed by Casazza and Christensen and appeared in [46]. It is considerably more complicated than (6.4), and we need some preparation before the proof.
Lemma 6.4.1 Let {efc}^_1 be an orthonormal basis for a finite dimensional Hilbert space Hn. Define
and
V 1 = 1
Then the following holds:
(i) {fk}k=i ™ a tyht frame for Hn with frame bound A = 1.
(ii) Assume that n > 2, and let {fkfriei be any subset of {A}^1 for which spanlfkfriei = Hn. Then, for an arbitrary ordering of the elements, {fk,}iei has basis constant greater than or equal to
iVn - 2-
Proof. To prove (i), let f G Hn and write
f = ^akek, where ak = {f,ek}. k=i
Letting P denote the orthogonal projection onto the unit vector Sfcsi e*>
Therefore
=	= i</,/n+i>i2.
130
6. Frames versus Riesz Bases
Also,
ll(i-n/ll2
Ek/,a>i2-
/c=l
Putting the two last results together, we obtain that
n+l
н/п2 = ii-p/ii2 + ii(i -p)/ii2 = Ek/’^i2-
/c=l
This proves (i). To prove (ii), we note that 52 ^_1 fk — 0, i.e., the vectors {fk}k=i are linearly dependent. Therefore any subset of the frame which spans 7Yn must contain fn+i and at least n — 1 of the terms {A}£=1. The basis constant for an arbitrary sequence is by its definition in (3.8) larger than or equal to the basis constant for any subsequence; thus, it is enough to prove the following
Claim: For any family of the type
{Л}кеди{п+1}, where Д C {1,2,..., n}, |Д| = n - 1,	(6.5)
there exists an index set Л С Д U {n + 1} and scalars	such
that
00
ECfcA
/сел
> -~y/n - 2
4
52 Ск^к
fc£AU{n+l}
(6.6)
Note that this takes care of the fact that {fk}ke&j{n+i} can be ordered in an arbitrary way.
Given a set A as in (6.5), let Ac = {^} denote its complement in {1,..., n}. Then, since 52/2= i fk — 0, we have
Ел
/сед
6.4 A frame which does not contain a basis 131
JJote that /п+1±Д for all к = 1,..., n. Therefore
E A
fceAu{n+i}
Ea
(6.7)
Let LeH’J denote the integer part of (see PaSe 419). For any subset Г C {1,2,..., n} with |Г| =	, we have,
1П < §
and |Г| >	- 1

therefore
Ea
We now consider a subset {fk}keA of {fn+i} U {fk}ke&- We choose A such that {fk}keA contains exactly of the elements of the set {fk}k=r Now, there are two possibilities.
Case 1 fn+1 £ {fk}keA-
Then as we saw in the estimate (6.8),
E A
1 /П
- 2V 2
132
6. Frames versus Riesz Bases
while by (6.7)
E a
fcGAu{n+l}
< V2.
Therefore
Ел
/сел
1 fn’ 1
2 V 2	y/2
E a
fceAu{n+i}
~Vn - 2
4
E A
fceAu{n+i}
Hence, (6.6) is satisfied.
Case 2 fn+i € {fk}keA-
Now, since /п+1±Д, к = l...,n, we have
Ел
кел
while we still have || £\еди{п+1} All < Thus (6.6) is again satisfied.
This completes the proof.	□
We are now ready to prove the existence of a tight frame which is norm bounded below and for which no subfamily is a Schauder basis. The exact meaning of this is that no matter how a subset of the frame is ordered, it is not a basis.
Theorem 6.4.2 In every separable infinite-dimensional Hilbert space H there exists a tight frame which is norm bounded below, and which does not contain a Schauder basis for H.
Proof. Since all infinite-dimensional separable Hilbert spaces are isometrically isomorphic, it is enough to construct an example in one particular Hilbert space. Let	be an orthonormal basis for a Hilbert space
AS and define an infinite collection of finite-dimensional vector spaces by
6.4 A frame which does not contain a basis
133
= span{ei}, Я 2 = зрап{е2,ез}, and in general
span s e (n — 1 )n .., e (n — 1 )n , o,..., e (n—1 )n .
L 2	' 1	2	' Z	2	~''L
Consider the direct sum
(00	\
n=l	/ ^2
as defined in (A.4). In each space Hn we construct the sequence as in Lemma 6.4.1, starting with the orthonormal basis
Specifically, given n 6 N,
We now show that {f^}^[,^==1 is a tight frame for H with frame bound A = 1. Write g G H as
9 = (Pi,P2, • • •), 9n E
We identify elements in a space Hn with their counterpart in H, i.e., we do not distinguish between f 6 7Yn and the sequence in H having f in the n-th entry and otherwise zero. Given n £ N it is clear that
{9, fk)n = {9n, for к = 1,..., n + 1.
It now follows from Lemma 6.4.1 that
OO n+l	00 n-f-1	00
EEiw£)hI2 = EEkw^j2 = Ei^iiL
n=l /c=l	n=l fc=l	n=l
= llpllli-
That is,	is a tight frame for H as claimed. Note that for n > 2
and 1 < к < n,
while
\\ш\
Л1 =Oand	= 1, VnGN.
134
6. Frames versus Riesz Bases
Removing // we thus obtain a tight frame which is norm-bounded below.
Now let {hk}(^L1 be any spanning subset of the frame	Xi ’ or‘
dered in an arbitrary way. The basis constant for {hk}(^=1 is by definition greater than or equal to the basis constant for any subsequence of {hk}(^=1 Now, for any n > 2, there exists N G N such that	contains a sub
sequence which spans 7Yn; this follows from the special construction oi 7Y as an orthogonal sum of the spaces 7Yn, and the choice of the fram< {fk }fcii’Xr By Lemma 6.4.1(ii) this implies that the basis constant for {^k}k=i is at least — 2; since n > 2 was arbitrary this implies that the basis constant for {hk}^^ is infinite. Thus, by Theorem 3.1.4, is not a Schauder basis for 3-L.	□
We note that a frame which is norm-bounded below satisfies inequalities, like
0 <	\\fk\\ < sup НДЦ < oo;
к	k
thus we have a slight variation of Theorem 6.4.2 (Exercise 6.3):
Corollary 6.4.3 In every separable infinite-dimensional Hilbert space there exists a normalized frame which does not contain a Schauder basis.
Vershynin has obtained an improvement of Theorem 6.4.2 [281]: there exists a frame {A}Xi which does not contain a basis with brackets, i.e., a subfamily {#n}Xi f°r which there exist numbers 1 < ni < П2... such that every f G H has a unique representation f = lim; J2Xi an%n- Bases with brackets only require the convergence of some special partial sums, and is thus a more general concept than a basis.
6.5 A moment problem
Let {AlfcLi be a sequence in a Hilbert space H, and let	6 £2(N).
It is natural to ask whether we can find f G H such that
VUN;	(6.9)
a problem of this type is called a moment problem. It is clear that there are cases where no solution exists: if for example Д = fj for some к j, a solution can only exist if = aj. More generally, if {fk}kLi is ^-dependent, we can find coefficients	(not all zero) such that J2Xi akfk = 0. If
for example a3 ф 0, then fj = —^2k^j ^fk', thus, (6.9) can only have a solution if aj is equal to the corresponding linear combination of {ak}k^j-
If the moment problem (6.9) has a solution, it is unique if and only if {A}Xi is complete (Exercise 6.4). We now present one more equivalent condition for a frame to be a Riesz basis; it is formulated in terms of the adjoint of the pre-frame operator T:
6.5 A moment problem 135
Theorem 6.5.1 Let {fk}^! be a frame forH. Then {fk}^! is a Riesz basis if and only if the range of the associated analysis operator T* equals £2(N).
proof. First assume that {fk}kL1 is a Riesz basis. Let {gkY^-t be the biorthogonal sequence. For	G /?2(N), Theorem 3.2.3 shows that
f	ak9k is well defined; furthermore,
Thus T* is surjective. On the other hand, if we assume that T* is surjective, = ^2(N), then Ar =	= {0}. Now the conclusion follows from
Theorem 6.1.1.	□
An alternative formulation of Theorem 6.5.1 is that a frame {fk}kLi is a Riesz basis if and only if the moment problem (6.9) has a solution for all {ak}<^L1 6 ^2(N). We note that Young proves a stronger result in [279]: for an arbitrary sequence {fk}kL1 in a Hilbert space, the range of the operator f {(/, fk)}kLi contains 4?2(N) if and only if there exists a constant A > 0 such that
n	n	2
/c=l	/c=l
for all finite sequences {cfcjjg.p A sequence {fk}kLi satisfying these conditions is called a Riesz-Fischer sequence.
For a frame which is not a Riesz basis, it follows from Theorem 6.5.1 that there exist sequences {ak}(^_1 6 /?2(N) such that (6.9) has no solution. But we can still ask for a best approximative solution’, here we have to give a precise definition of how we want to approximate the given sequence	Since we are working with ^2(N), the natural question
is whether we can find an element in 7Y which minimizes the functional f Ebl \ak ~ (fi A)I2; the answer turns out to be yes:
Theorem 6.5.2 Let {fk}kL1 be a frame forH, and let {а^}^ G ^2(N). Then there exists a unique vector in TL which minimizes the functional
00
/c=l
this vector is f = JZfcli akS~1fk-
Proof. By Proposition 5.3.6, the orthogonal projection of a sequence {q}£1i G ^2(N) onto the range of T* is given by
f 00	1 °°
P{ck}?=1 = ] CbS-'fk, fj) >	•	(6.10)
I k=l	Jj-!
136
6. Frames versus Riesz Bases
The functional f	\ak — {f, A)|2 is minimized when
which is the case for f =	akS~1 fk', by completeness of {/к}^ the
minimizer is unique.	□
The above use of language might be slightly confusing: the mapping f	\ak ~ {f^ fk)\2 is not linear, but still it is traditionally called a
functional.
Corollary 6.5.3 Assume that {fk}kLi is a Riesz basis for TL, and let {ufc}^-i E ^2(N). Then the moment problem (6.9) has a unique solution, which is given by
oo	oo / oo	\
f = '^akS-1fk = £	Л;
j=l \/c=l	/
here (T*T)j £ is the jk-th entry in the matrix representation for T*T with respect to the canonical basis.
Proof. Since we know that the moment problem has a solution f when {fk}kLi is a Riesz basis, the representation of f via S-1 follows from Theorem 6.5.2. By Theorem A.7.3 the solution can also be expressed via the pseudo-inverse of T*:
f =
= T(T*T')~l{ak}%L1
OO / OO	\
= E E(rT)>
j=l \k=l	/
□
6.6 Exercises
6.1	Give an example of a Riesz frame which is not a near-Riesz basis.
6.2	Find a frame which contains a Schauder basis which is not a Riesz basis. (Hint: use Example 5.1.4).
6.3	Prove Corollary 6.4.3.
6.4	Let {fk}^} be a sequence in TL, and let {afc}^ G ^2(N). Prove that	is complete in TL if and only if (6.9) has at most one
solution.
7
Frames of Translates
The previous chapters have concentrated on general frame theory. We have only seen a few concrete frames, and most of them were constructed via manipulations on an orthonormal basis for an arbitrary separable Hilbert space. An advantage of this approach is that we obtain universal constructions, valid in all Hilbert spaces.
In order to apply frames in e.g., signal processing it is necessary to be more specific and construct frames in concrete Hilbert spaces consisting of functions. This will be the central theme in the following chapters, where we show how to do this in L2(K) and subspaces hereof. All frames will be coherent, i.e., all elements in the frame have a common structure. The exact meaning of this will become clear as soon as we define the frames, but the idea is that each element in the frame	appears by the action of an
operator (belonging to a special class) on a single element f in the Hilbert space. This feature is essential for applications: it simplifies manipulations on the frame, and makes it easier to store information about the frame.
In this chapter we consider the case where the operators act by translation. That is, we consider families of the form {ф(- — Afc)}fcez, where is a sequence in R and ф 6 L2(K). Using the translation operators defined in Section 2.5 we can write {ф(- — Afc)}fcez — {Т\кФ}ке%-Recall also the modulation operator Exk from Section 2.5: since translation of a function corresponds to modulation of its Fourier transform, i.e., ЕТхкф = Е_хкЕф, frames of translates are closely related to frames consisting of complex exponentials {e1AfcI}/i;€z; a section is devoted to this type of frames, too.
138
7. Frames of Translates
Frames of translates are natural examples of frame sequences. In fact’ we will prove that {Тхкф}ке% most can be a frame for a pure subspace of L2(IR). In the next chapters we will see that sequences of translates (respectively, complex exponentials) are of fundamental importance also for construction of frames in L2(IR).
This short introduction already indicates that the translation operators and modulation operators will play a central role, together with the Fourier transform.
This chapter brings us close to the origin of frames: historically, Duffin and Schaeffer introduced frames in the context of sequences of complex exponential functions. Their paper [121] from 1952 contains the general definition of frames, but the core subject is nonharmonic Fourier series. Young has given an outstanding presentation of complex exponentials and nonharmonic Fourier series in his book [279]. For this reason we will concentrate on results which are directly related to frames, and mainly discuss those that appeared after the first edition of [279] from 1980. Several results will be presented without proofs, and the reader who knows [279] will understand why: a deeper analysis requires advanced complex analysis, and would bring us too far away from the main theme.
7.1 Sequences in
We begin by defining some types of sequences in .
Definition 7.1.1 Let I be a countable index set and {Xk}kei a sequence in IRd. We say that
(i)	A point Л G is an accumulation point for {Xk}kei if every open ball in centered at X contains infinitely many Xk.
(ii)	{Xk}kei is separated if inf|Aj — A&| > 0; a constant 5 > 0 such that |Aj — Л&| > 5 for all j к is called a separation constant.
(iii)	{Xk}kei is relatively separated if it is a finite union of separated sequences.
A relatively separated sequence can repeat the same point N times for some N G N, but it can not have an accumulation point.
Example 7.1.2
(i) The sequence {^}fcez\{o} has zero as accumulation point.
(ii) The sequence {/с, к 4- ртщНег has no accumulation point and is not separated. However, it is relatively separated.	□
7.1 Sequences in Rd 139
* Several characterizations of relatively separated sequences are known, probably the most important one is in terms of the upper Beurling density, which we now introduce. For the sake of short notation, denote the given sequence by A = {Xk}ke%- For x G and h > 0 we let Qh(x) denote the half-open cube in Rd centered at x and with side lengths h, i.e.,
d
Qh(%) =	— h/2,Xj 4- h./2[, where x = (xi,... ,xt).
Note that {Qh(hn)}neZd is a disjoint cover of Rd for any h > 0. Let v+(h) and v~(h) denote the largest and smallest numbers of points from A that lie in any cube Qh(x), i.e.,
v+(h) = sup Й (Л П Qh(x)), v~(h) = inf (Л A Qh(x)) .
The upper and lower Beurling densities of A are now defined as
P+(A) = limsup У respectively, P~(A) — liminf . (7.1) Д_____________>oq П'	h
In case P+(A) = P~(A), we say that A has uniform Beurling density
P(A) = Р+(Л) = /Г(Л).
Lemma 7.1.3 Let A = {Xk}ke% be a sequence in Rd. Then the following are equivalent:
(i)	D+(A) < oo.
(ii)	A is relatively separated.
(Hi) For some (and therefore every) h > 0, there is a natural number Nh such that each cube Qh(hn), n 6 Zd, contains at most Nh points from X, i.e.,
sup (A A Qh(hn)) < oo. n£Zd
Proof. (i)=> (iii). If (i) is satisfied, there exists a constant N such that for all sufficiently large h > 0,
hd
this gives (iii) for large values of h (as a consequence, it holds for all h > 0).
(iii)	=> (ii). Let h > 0 be chosen such that (iii) is satisfied. We will show explicitly how A can be split into a number of h-separated sequences. Let ci,..., e2d denote the vertices of the unit cube [0, l]d, and consider the sets
Z3 = (2Z)d + ej, j = l,...,2d.
140
7. Frames of Translates
Note that is the disjoint union of the sets	Since
{Qh{hn)}neZd is a disjoint cover of this implies that is the disjoint union of the sets
Bj= U Qh(hn), j = 1,... ,2d.
nEZ3
Let us analyze the cubes Qh(hn) which form a given set Bj. If we consider some m,n 6 Zj with m n, the distance between the cubes Qhlhn) and Qh(hm) is at least h, i.e., the distance between arbitrary elements in Qh(hn) and Qh(hm) is at least h. By the assumption in (iii), each cube Qh(hn) contains at most elements from A, i.e., t] (A A Qh(hn)) < Nh; since
AnBj = [J (ЛПфДЛп)), neZj
it follows that A A Bj can be split into sets which are h-separated. Therefore A can be split into 2dNh sequences which are h-separated.
(ii)=>(i). Assume that (ii) is satisfied; by definition, this means that we can choose a partition of A into a finite number of sequences, say. Ai,...,Ar, such that each sequence A^ is separated with separation constant, say, 6k < 1. Let
x • M	M
о := mm < —=,..., —> .
[Ъ/d	2y/d)
The maximal distance between points in any cube Q§(x) is min{Ji,..., 5r}, so the cube contains at most one point from each sequence A^, and therefore at most r points from A. Thus, if h is any positive number, then Qhs(x) contains at most r(h 4- l)d elements from A. Via the definition of D+(A.) this implies that
P+(A) = limsup V- ;y -h-4-ОО hd
i/+(M) hm sup -y-h-+oo (h6)d
lim sup h—>oo
r(h + l)d (h5)d
r
oo.
□
7.2 Frames of translates
We are now ready to discuss the frame properties of sequences consisting of translates of a function ф e L2(R). The main question is: which conditions on a real sequence {Afc}kGZ and a function ф e L2(R) will imply that {Tx*:	is a frame?
7.2 Frames of translates 141
Later in this chapter we prove Theorem 7.4.1, which shows that {Тхкф}ке1 can not be a frame for all of L2(K). However, frame sequences of the form {Т\кф}кег exist. With a slight abuse of the language we will usually skip the word “sequence” and refer to {7Afc0}fcGz as a frame of translates.
The theory for frames of translates is far from being fully developed, and for a given sequence	and a function ф it is often difficult to find out
whether {Т\кф}ке% is a frame or not. The interplay between {Xk}ke% and Ф is complicated: certain conditions on the density of {A^J^ez are necessary for {Т\кф}ке2 to be a frame (Exercise 7.1 or Theorem 7.4.1), but if they are satisfied the final answer still depends heavily on the choice of ф.
We begin with the special case where A^ = kb for some b > 0. Because the points {kb}kez are equidistantly distributed, a frame {Ткьф}ье% is sometimes called a regular frame of translates in contrast to the irregular frames {ТХкф}ке2-
Let ф 6 L2(IR). Our first goal is to prove a result by Benedetto and Li [22], which shows that the frame properties for {Ткьф}ке% can be completely described in terms of the function
Ф : IR -> IR, Ф(7) = ф
fcez
(7.2)
Note that Ф is 1-periodic and that
о
i.e., Ф 6 L1(0,1). We begin with some lemmas, which will we used
repeatedly.
Lemma 7.2.1 Let ф 6 L2(K) and assume that {Ткф}ке^ is a Bessel sequence. Let {ck}kez E £2(Z). Then ^кег^ТкФ converges in L2(K) and ^2kEZckE^k converges ml2(0,l), and
^^скТкф = (^2скЕ_к\ф.	(7.3)
fcez	\fcez /
Proof. That ^ке2скТкФ and ^ke%ckE-k converge as described follows from Theorem 3.2.3, so we only have to prove (7.3). First we note that
Е^скТкф = ^скЕТкф = ^(скЕ_кф), k<EZ	kEZ	kEZ
142
7. Frames of Translates
where the series on the right-hand side converges in L2(R). We have to prove that
^(скЕ_кф) = I ^ckE_k j ф. kez	\fcez /
Now,
fcez
L2(R)
^Г(скЕ_кф) - скЕ_кф
kez	|fc|<N
^2 СкЕ-кф - I У2СкЕ~к 1 Ф
|fc|<N	\fc£Z /
(7.4)
(7.5)
The term (7.4) converges to zero for N -» oo, so we only have to estimate (7.5). Since ^кЕ2скЕ_к is 1-periodic,
L2(R)
/2
\ 1/2
fcez	/
скЕ—к СкЕ_к
|fc|<N	fcez
here В denotes a Bessel bound for {Ткф}кЕ^- The last term converges to 0 as N —> oo, and the proof is completed.	□
Lemma 7.2.2 Let a > 0 be given, and f : К -» C be a bounded a-periodic function. Then, for g 6 ZJ (1R),
[ f(x)g(x)dx= f f(x}'^g(x-ka)dx.
. -o°	fcez
7.2 Frames of translates 143
proof. We first show that
l/WI J2l5(z-fca)|da; < oo.	(7.6)
kez
For positive functions, sums and integrals can be interchanged, so
[ 1/(ж)1 ^2 |p(s - fca)|dx = 52/ l/MI li'C1 ~ ka)\dx = (*)•
J °	kez	кеяУ0
Using that f is assumed to be a-periodic,
(*) = 52 [ \f(x - ka)\ \g(x - ka)\dx = f |/(x)| |p(x)|dx,
«/-oo
which is finite because f is bounded and g G LX(1FL). This proves (7.6); the result now follows from Lebesgue’ domimated convergence theorem. □
We are now ready for the announced characterization of frame properties for {ТкЬф}кЕ%-
Theorem 7.2.3 Let ф 6 L2(№) and b > 0 be given. For any А, В > 0, the following characterizations hold:
(i)	{Ткьф}кЕ% is a Bessel sequence with bound В if and only if
Ф(7) < ЬВ, a.e. у G [0,1].
(ii)	{Ткьф}ке2 is an orthonormal sequence if and only if
Ф(7) = b, a.e. у G [0,1].
(iii)	{Ткъф}ке% is a Riesz sequence with bounds A, В if and only if
ЬА < Ф(7) < bB, a.e. у G [0,1].
(iv)	{Ткьф}кЕ1 is a frame sequence with bounds A, В if and only if
ЬА < Ф(7) < bB, a.e. 7 [0,1] \ N,
where N = {7 G [0,1] : Ф(7) = 0} .
Proof. To prove (i) we note that without any Bessel assumption, the pre-frame operator
T : {ck}kez -» У^скТкьф
kez
is well defined as a map from all finite sequences in £2(Z) to L2(IR). Given a finite sequence {ck}k<=z we consider the trigonometric polynomial in L2(0,1) given by /(7) =	cke~2niky. Then, using that the Fourier
144
7. Frames of Translates
transform is unitary and the commutator relation ЕТкь = E-kbE,
l|r{cfc}fcez||2
CkTkb<f) kez
ЕУ^скТкьФ kez
cke~2*ikb^)
2
d'y
У |/(^7)|2 p(?)| d'i
Via Lemma 7.2.2 we can continue with
||r{cfc}kez||2 = | fl/(7)|2 £ 4Ч2) 2rf7 й Jo kez V ° /
= | /'1|/(7)|2Ф(7М7.	(7.7)
0 Jo
КФ(7) < bB for a.e. 7, it follows that
nmwi2< f i/(7)i2^7 = E ы2;
kez
via Exercise 3.9 this implies that {ТкЬф}к& is a Bessel sequence with bound B. To prove the opposite implication in (i) we note that if {Ткьф}ке% is a Bessel sequence, then the calculations leading to (7.7) holds for all {ckjkez £ £2(Z). Denoting the Bessel bound by В we conclude from (7.7) that
|	< sEn2
° Jo	keZ
= В I |/(7)|2rf7, Vbhez e ^(Z).
Jo
This implies that $(7) < bB for a.e. 7 and concludes the proof of (i).
We now prove (iv) via Lemma 5.5.4. Since we have proved (i) we will assume that {Ткъф}ке% is a Bessel sequence, and concentrate our analysis on the lower bounds. Consequently the equality
IlTfc/JfcgzH = i f |/(7)|2 Ф(7)с?7	(7.^
7.2 Frames of translates 145
now holds for all sequences {ck}kez G /?2(Z); it implies that
Vt = | {ск}кег 6 £2(Z) I £ ске~2^ = 0 on [0,1] \ И .	(7.9)
I	kez	)
For arbitrary sequences {ck}kGz, {dfcbez € ^2(Z), the fact that {e2lrzkx}kez is an orthonormal basis for L2(0,1) implies that
({ahez,№WHz) = 0	= 0;
\kez	kez	/ £2(о,1)
dt follows that (Exercise 7.2)
= j Mkez &	| 52 c*e"2"fc7 = o on NI.	(7.10)
\	k£Z	J
So for {ck}kez € Л//,
kez
ске~2п^
dy =
2
c?7;
2
using (7.8), the left-hand condition in (5.15) in Lemma 5.5.4 is therefore equivalent to
2
A f ^cke~2^ dy ^[°,1]\N fceZ
2
$(7)d7, V{c*}*6Z e A#.
This, in turn, is equivalent to (Exercise 7.2)
bA < $(7) a.e. у 6 [0,1] \ N.
(7.11)
This proves (iv). For the rest of the proof, recall that the Riesz bounds and the frame bounds coincide for Riesz sequences. By Theorem 3.6.6, {Tkb<l)}kez is a Riesz sequence if and only if the inequalies (5.15) hold for all {ckjkez £ ^2(^); this is the case if and only if Л/т = {0}, i.e., by (7.9), if and only if N is a null-set; this gives (iii). (ii) now follows from Proposition 3.4.8.	□
Example 7.2.4 Let a e]0, |[ and define ф 6 L2(IR) via its Fourier transform as <£(7) = Х[-а,а[(?)- Take b = 1. Then, for 7 G [— |, |[,
$(?) = Х[-а,а[(?)-
Theorem 7.2.3 shows that {Ткф}кЕ% is a frame sequence with frame bounds A = В = 1. {Ткф}кЕ^ is not a Riesz sequence, i.e., it is an example of an over complete frame of translates.	□
146
7. Frames of Translates
As another application of Theorem 7.2.3 we now consider the n-th order B-spline defined in Appendix A.9. We claimed in Lemma 3.6.10 that {TkBn}kEz is a Riesz sequence. We already noticed that the result holds for n = 1; in fact, {T^BiJ^ez is an orthonormal system, which by Theorem 7.2.3 implies that
E |^i(7 + fc)| - 1, a.e.-у.
fcez
Since Bn(y) = (Bi(y))n by Corollary A.9.2 and |Bi(7)! < 1 for all 7, it immediately follows that
E|-Bn(7 + fc)| < У2|-В1(7 + ^)| =1, a.e. у.
k€Z	keZ
Thus {TkBn}kEz is a Bessel sequence. On the other hand, again by Corollary A.9.2,
E|B„(7 + fc)|2> inf |Bn(7)|2=
and the result follows from Theorem 7.2.3.
Exercise 5.12 shows that if {ek}kEz is an orthonormal basis or a Riesz basis, then {ek 4- ek+i}kez can not be a frame. Via Theorem 7.2.3 we can prove that the situation changes if we allow {ek}kEz to be a frame:
Proposition 7.2.5 Let ф € L2(JR) and assume that {Ткф}кЕх is a frame for V := зрап{Ткф}к^- Then the following are equivalent:
(i) {Ткф 4- Tfc+1^}fcGz is a frame for V.
(ii) Ф — 0 on a neighborhood of у =
Proof. A slight modification of the argument in Example 5.4.6 gives that V = span{7T^ 4- Тк+1ф]к^- Letting ф := ф 4- Tip, we can write Ткф 4- Тк+1ф = Ткф. Via the Fourier transform, Тф = (1 4- Е-1)ф, so
Ф(7) == £ 1^(7 + <
к £Z
= E|l+E_1(7 + fc)|2 Й7 + < kez
= |l+e-2^|2y>(7 + <-
к G Z
Now the result follows from Theorem 7.2.3.	□
Concrete examples of frame sequences {Ткф}кег satisfying condition (ii) i can be found via Example 7.2.4.
7.3 Frames of integer-translates
147
7.3 Frames of integer-translates
Frames of translations with b = 1 play an important role in the theory for frame multiresolution analysis, as we will see in Chapter 13. For this reason we now consider this case more closely. Given ф e L2(JR) we let as before
$(7) = FW7 + C	(7-12)
k£“Z
Recall the definition of the dilation operator Da in Section 2.5. A scaling will transfer all results for frames {Ткф}ке2 to results for {Ткафа}ке%, where фа is a scaled version of ф:
Lemma 7.3.1 Let ф € L2(JR) and b > 0 be given. Assume that {ТкьФ}ке% is a frame sequence. Given a > 0, let фа := Баф. Then {Ткьафа}ке% is a frame sequence with the same frame bounds as {Ткьф}ке2-
Proof. We just notice that by the commutator relations,
DaTkb — TkbaDa]
the rest follows from Lemma 5.3.3.	□
We already now notice that membership of span{Tfc^}fcGz can be characterized in terms of the Fourier transform when {Ткф}ке% is a frame sequence. We will prove a more general version in Lemma 13.2.1.
Lemma 7.3.2 Assume that ф e L2(JR) and that {Ткф}ке% is a frame sequence. Then a function f E L2(IR) belongs to зрап{Ткф}ке2 if and only if there exists a 1-periodic function F whose restriction to [0,1[ belongs to L2(0,1), such that
f = F$.
In order to apply Theorem 7.2.3 it is essential to be able to control the function Ф. Frequently, it is useful to express Ф in terms of its Fourier series:
Lemma 7.3.3 Let ф € L2(JR). Then the Fourier coefficients for the function Ф e //(ОД) with respect to the orthonormal basis {e2lrikx}kez are
Ск = [ ф(х)ф(х 4- k)dx, ktzTj.	(7.13)
J —oo
148
7. Frames of Translates
Proof. Using the 1-periodicity of the modulation operator Ek, the Fourier coefficients for Ф can be expressed by
Ck = f1 $(7)e-2”^d7
Jo
Jo nez
Via Lebesgue’s dominated convergence theorem, we can interchange the sum and the integral; thus,
<7 = Г W7)|V2”^7
J — oo
= <Ф,Екф}
= <Ф,Т-кф).
Corollary 7.3.4 Assume that ф € L2(JR) is real-valued and has compact support. Then Ф is a trigonometric polynomial of the form
N
Ф(-у) = cq 4- 2 Ck cos(2ttA;7) for some N G N,
k=l
where {q}£L0 are Fourier coefficients for Ф. In particular, {Tk<t>}k€L can not be an overcomplete frame sequence.
Proof. Let {ckjkez be the Fourier coefficients for Ф. Via Lemma 7.3.3, the assumption that ф is real-valued implies that q = c~k for all к 6 Z. Due to the compact support of ф, there is an N E N such that Ck = 0 if |/c| > N. Thus, Ф is a trigonometric polynomial; expressing Ф via its Fourier series,
ф(7) = E
|k|<N
N
= co + E	+ в-2"*7)
fc=l
N
= cp + 2 Ck cos(2tfA;7).
k=i
Since Ф is continuous, Theorem 7.2.3 gives the rest.
7.3 Frames of integer-translates 149
Example 7.3.5 Let ф = X[-i,2[- By Lemma 7.3.3, the Fourier coefficients for Ф are
' 3 if к = 0, 2 if к = ±1, 1 if к = ±2, 0 otherwise.
X.
Thus, by Corollary 7.3.4,
Ф(7) = 3 4-4cos(2tt7) + 2 cos(4tt7).
Note that Ф is continuous and that Ф has two isolated zero’s on [0,1[: ф(7) = 0 for 7 = | and for 7 = |. By Theorem 7.2.3 it follows that {ТкФ}ке2 is a frame sequence.	□
Without referring to the Fourier transform it is difficult to find a function ф such that {Ткф}кЕ1 is an overcomplete frame sequence; already Corollary 7.3.4 shows that ф can not be a real-valued function with compact support. Other restrictions are given in the following proposition:
Proposition 7.3.6 Let ф E L2(JR) and assume that {Ткф}ке% is an overcomplete frame sequence. Then the following holds:
(i) The function Ф is discontinuous.
(ii) Either ф LX(R) or there is no constant C > 0 for which
1 \1/2
*
< C (
(7-14)
Proof, (i) follows directly from Theorem 7.2.3. For the proof of (ii), assume that an estimate of the type (7.14) is available, and let 7 e [0,1[. Then, for all N e N,
Ф(7)- E 1^(7 + <
= E 1^(7+fc)i2
|*|>Л
< г2 V 1
’ pfev 1 + 17 +
2C\^-
Since n	—> 0 as N -> 00, this shows that the series
.	E 1^(7+*oi2
к
150
7. Frames of Translates
is uniformly convergent. Thus, if ф was continuous, then Ф would be con tinuous as a uniform limit of continuous functions. This would contradict (i), so ф can not be continuous, and therefore ф £ L1(JR).	□
The interpretation of Proposition 7.3.6 is that a function ф generating an overcomplete frame sequence {ПФ}ке% has bad time-frequency local-, ization: either ф does not decay fast, or its Fourier transform has slow decay.
We now turn the focus to the dual of a frame of translates. Assuming" that {ПФ}ке% is a frame for its closed linear span V, the frame operator!
s-. V^V, 3/ = ^(/,Ткф)Ткф
k€Z
is invertible. It is important to notice that S and S-1 commute with integer translation:
Lemma 7.3.7 Let ф E L2(JR) and assume that {ПФ}ке% a frame for its closed linear span V. Then
STk = TkS and S-'Tk = US'1 on V, V/c E Z.
Proof. Given f E V and к E Z, we have
STkf = ^ТкГТк'ф}Тк,ф
fc'GZ
= у^(/,тк,_кф}тк,ф.
k'EZ
Replacing the summation index к' Ъу к' + к gives
srkf = y^f'Tki,t,')Tk'+k(t> k'ez
= TkSf.
The second part of the result follows from here.	□,
By the general definition, the canonical dual frame associated with {Пф}ке% is given by {S~1T^}kez- Using Lemma 7.3.7 we see that
{S~lTkv}k^= {TkS~4}kez-
This result is very useful for calculation of the dual frame. In order to’ find {3~1Ткф}ке2 we would have to compute the action of S-1 on the infinite family of functions {ПФ}ке%- On the other hand, calculation of {ПЗ-1 Ф}ке% only requires that we find the rest of the functions in the family are obtained by translation. This is certainly a simplification, but we are still left with the question of finding 3~1ф. The problem is that
7.3 Frames of integer-translates
151
5 is an operator on V, which is usually infinite-dimensional; theoretically, we know that S is invertible, but this is different from being able to find the inverse explicitly! For general frames we return to this problem in Chapter 16. In the present context we are able to find 3~1ф in terms of its Fourier transform, see Proposition 7.3.8.
proposition 7.3.8 Let ф E L2(JR) and assume that {Тьф}ье% a frame for its closed linear span V, with frame operator S. Let
D := {7 E 1R : Ф(у) 0},
and define the function 0 via its Fourier transform by
0(7) :=
Then 0 = 3~1ф.
proof. The function
if 7 E P, if P
ify^D, if g D.
(7-15)
is 1-periodic, and its restriction to ]0,1[ belongs to L2(0,1). Thus, Lemma 7.3.2 shows that the function 0 defined by (7.15) belongs to V. Using the definition of the frame operator, properties of the Fourier transform, and Lemma 7.2.1, we have
FS0 =
= ^^ПФ)^Ткф
ТУ,Е_кф)Е_к]ф.	(7.16)
fcez	/
Now, using that the exponentials 7 -> е27ггА:7 are 1-periodic, and the definition of 0,
{в,Е-кф) = fi
J —00
= [ IL + n)fa + n')e27Ti<k+n,'!\ d-t
=	/ XDn[o,i[(7)£-fc(7)rf7,
Jo
152
7. Frames of Translates
which is the —/с-th Fourier coefficient for the function XDn[o,i[ *п £2(0.1). Therefore
= XDn[o,i[ on [0,1]-
fcez
Since xd is 1-periodic, it follows that
^Г{0,Е-кф}Е_к =xd on R
fcez
Noting that xd(t) 0 0 if ^(7) 0 0, (7.16) now implies that
FSO = хоф = Ф-
Therefore S0 = ф and since S is an invertible operator on V, the proof is over.	□
For the case where an orthonormal basis is preferred, Daubechies proved in [106] that any Riesz sequence {Ткф}кЕ% can be transferred to an orthonormal sequence which spans the same space. The orthonormalization trick applied to a frame sequence will lead to a tight frame for the same space:
Proposition 7.3.9 Let ф G L2(IR), and assume that {Ткф}кЕ% is a frame sequence. Define the function ф$ via its Fourier transform by
(7.17)
l 0	if 0(7) = 0.
Then {Tfc^}kez is a tight frame sequence, and span{T^}kez = зрап{Ткф}кег-
If {Ткф}ке% is a Riesz sequence, then {Ткф$}кЕ2 is an orthonormal sequence.
Proof. The reader can check that ф$ is well defined, i.e., that $(7) 0 0 if <£(7) 0 0. Define
$’(7) := ^2 7^(7 + fc)|2 •
ke%
By Theorem 7.2.3 we want to prove that Фй is bounded above and below away from its zero set. Let 7 G [0,1[. If <£(7 + к) = 0 for all к G Z, then Ф#(7) = 0, so we now assume that <£(7 4- к) 0 for some к e Z. Then for all к G Z we have that
0 7^ Ф(-7 4- к) = Ф(-7)-
7.4 Irregular frames of translates 153
therefore the definition of Фй gives that
Ф»(7)	=
= 5к2'*(7+<
= 1 a.e..
Since Ф# only assumes the values zero and 1, Theorem 7.2.3 implies that is a tight frame. In the special case where {Ткф}kgz is a Riesz sequence, we can for a.e. 7 E [0,1] find к E Z such that ф(у 4- к) 0; thus фй = 1 a.e., i.e., {Ткф^}ке% is an orthonormal sequence.
In order to prove that {Ткф*}кея spans the same space as {Ткф}ке% we note that
§рап{Т*0#}кб2 = <j ^скТк<^ | {cfc}kez € ^2(Z) >. taez	J
Taking the Fourier transform of the functions in this space and letting
F( } f Ф~1/2(7) if <^(7) / 0, J ' 11 if ^(7) = 0
yields
1	I {ск}к& e f2(Z) I = j F^CkE-кф I {c*}*6Z € £2(Z) I
uez	J I fcez	)
The function F is bounded above and below, so
I {ojfcez € ^2(Z) > = < CkE-кф | {ck}kez € £2(%) > ;
к &GZ	J	\kGZ	J
this final space equals the space of Fourier transforms of the functions in span{Tk^}kez- Thus Рзрап{Ткф^}ке2 = FspaF{Tk0}kEz, and the result follows.	□
7.4 Irregular frames of translates
We now return to the general case, where {A/Jkez is an arbitrary sequence in R. We have already noted that it is difficult to prove whether {Т\кф}кЕ% is a frame or not, except under very special conditions on ф (Exercise 7.4).
The first part of the following theorem shows that if we want {Т\кф}ке% to be a frame sequence, then we have to assume that {Akjkez is relatively separated; otherwise {Т\кф}ке% can not be a Bessel sequence in L2(R) for any function ф e L2(R) \ {0}. The second part shows that {Xk}kEZ being
154
7. Frames of Translates
relatively separated excludes {Т\кф}к£% from having a lower frame bouni in L2(R). Put together, the conclusion is that {Т\кф}к£% never can be a frame for all of L2 (R) (but it can very well be a frame sequence). The result is due to Christensen, Deng, and Heil [77].
Theorem 7.4.1 Let A = {Xk}kez be a sequence in R and ф 6 L2(R)\{0}. Then the following holds:
(i) If {Т\кф}ке% is a Bessel sequence, then D+(A) < oo.
(ii) If {Т\кф}ке2 satisfies the lower frame condition in L2(R), then D+(A) =oo.
In particular, {Т\кф}к£% can at most be a frame for a proper subspace of L2(R).
Proof. Recall from Lemma 7.1.3 that D+(A) < oo is equivalent to A being relatively separated. For the proof of (i), we assume that A is not relatively separated; we have to prove that then {Т\кф}ке2 is not a Bessel sequence. Consider the function x -» {ф, Тхф), x € R. Since the function is continuous by Lemma 2.5.1 and non-zero for x = 0, there exists an interval ] — h, h[, h > 0, such that
M:= inf'	\(ф,Тхф)\ > 0.
— h,h[
Consider an arbitrary TV G N. By Lemma 7.1.3, there exists an interval ]a — h, a 4- h[, a € R, which contains at least N elements from the sequence A. Now, letting
A-N •— {к E Tj ; Xk — h, a -{	= {к E Z : Xk — a G] — h, /i[}-,
we have
£|<T^,TAfc< > £ \(таф,тХкф}\2 k&L
=	£ |^,TAt_a<
ke/^N
> w2
Since N € N was arbitrary, it follows that {Т\кф}ке% is not a Bessel sequence in L2(R). This proves (i).
For the proof of (ii) we assume that D+(A) < oo; thus, we have to prove that {Т\кф}ке% does not satisfy the lower frame condition for any A > 0 By Lemma 7.1.3, {A/Jajgz is a finite union of separated sets, i.e., we can
7.4 Irregular frames of translates
155
Bwrite s = |J {Afc}kei3,
where each set {Xk}kei3 is separated. Choose a constant 6 > 0, which is a separation constant for each sequence {Xk}kei3, 3' — 1» • • •>s> and consider e]0,5/2[. With I := [—h,h],
^\(хьТХкф)\2 = feez	j=i kei3
< EEllX/112 IIx/Ta^II2. (7.18) j=i kei3
|ty the choice of Z, the intervals {I — Xk}kei3 are disjoint; by defining
Aj := |J(/-Afc), kei3
we have
E iix/twii2 = E f w* -	= f
kei,	kei,Jl
£hus, via (7.18),
Екхл< imi2 E [ fcez	j=i
An application of Lebesgue’s dominated convergence theorem shows that for each fixed j = 1,..., s,
/ |ф(х)|2drr -> 0 as h -> 0;
Ja3
thus, {Т\кф}ке2 does not have a lower frame bound in L2(1R).	□
A more general result can be found in [77]: no union of arbitrary translates of a finite collection of functions gi,..., дм can be a frame for Z2(JR). The proof is almost identical to the above proof, only the notation is more involved. As a consequence of this result, for no function g 6 Z2(]R) and no constants a, b > 0 can a collection of functions of the form {TnaEmbg}nez,m=i,...,M be a frame for Z2(1R). However, frames of the type {ТпаЕтьд}т,пе% exist; they will be the topic of Chapters 8-9.
Proposition 7.4.2 Assume that {Xk}kez is a sequence for which X^ ± Xj for к j. If ф E L2(JR) \ {0}, then the functions {Т\кф}ке% are linearly independent.
156
7. Frames of Translates
Proof. Let T c Z be a finite set, and assume that for some coefficients {QjfcGJF,
£CfcT\^ = o. k^F
Via the Fourier transform,
СкЕ-хкФ = o.	(7.19)
fcey
Choose a bounded non-empty interval I on which ф is not identically zero. Assume that not all coefficients {ск}ке^ are zero; then the function 7 i->	ckE_ xk (7) is only zero for finitely many 7 E /, and then
скЕ-хкф is not the zero function. This contradicts (7.19); we conclude that Ck = 0 for all к E F, and that the functions {Т\кф}ке% are linearly independent.	□
7.5 The sampling problem
A short and not yet precise formulation of the sampling problem is: how can we find a function f : К -> C if we only know a countable set of function values {/(Afc)}^^/? Formulated this way the problem is ill-posed: there are infinitely many functions that take the same prescribed values on a given countable set, so we need to impose some condition on the function f for the problem to make sense. Traditionally, this is done by requiring f to belong to a certain function space. A classical case where this approach is fruitful is known as Shannon’s Sampling Theorem. It uses the sine-Junction, which is defined by
sin(7ra:)
7ГГЕ
1
if x 7^ 0, if x = 0.
Theorem 7.5.1 Assume that f € L2(R) and that f has support in [—1/2,1/2]. Then f can be recovered from the samples {f(k)}ke% via
f(x) — f(k)sinc(x — k).
ke%
We will give a short description of sampling in shift-invariant spaces and refer to the paper [3] by Aldroubi and Grochenig for more information and? further references.
The starting point is to consider a closed subspace H C L2(JR) consisting of continuous functions. We assume that for each x G IR, the point evaluation
f №)
7.6 Frames of exponentials
157
jS a continuous linear functional on in this case H is called a reproducing kernel Hilbert space. By Riesz’ representation theorem, there exists for each $ ёН & unique element Kx 6 H such that
f(x) = {f,Kx),VfeH.	(7.20)
The set of functions	is called the reproducing kernel.
A sequence	С К is a set of sampling for H if there exists
instants А, В > 0 such that
AIUII2 < £|Ш2 < В ll/ll2, V/eH;
fcez
by (7.20) this is equivalent to {K\k}kei being a frame for H.
As an example of a reproducing kernel Hilbert space, one can take a shift-invariant space
Н:=1УскТкф : {cfc}fc6Z g £2(Z) Lgz
where ф f= L2(R) is a continuous function for which
12 11^ Х[кл+1[||оо < oo. fcez
We return to this class of functions in Section 8.5. For now, we assume further that {Tfc^jfcez is a Riesz basis for H. Lemma 7.3.7 shows that the dual Riesz basis has the form {Ткф}ке%, where ф — 3~1ф. Now, if {Afcjfcez is a set of sampling for H, one can prove (cf. [3]) that the frame {K\k}kE% is given by
K\k = ^2	~ пУГпФ-
nE'Z
This describes how the frame elements can be found as linear combinations of the elements in the Riesz basis {Тпф}пЕ% and connects to the theme in Proposition 5.5.8.
7.6 Frames of exponentials
Recall that the complex exponential functions	constitute an
orthonormal basis for L2(-7r,7r). Thus {elkx}kez is a frame for L2(-7r,7r) with bounds A — В — 2тг. More generally, given an interval I C 1R and a real sequence {A/J^z, a frame for L2(Z) of the form {егХкХ}ье2 is called a frame of exponentials, or a Fourier frame. Note that the exponentials are not square integrable on an unbounded interval, so we necessarily have \I\ < oo. An expansion
/(^) = ^2c*eiAfc:r
158
7. Frames of Translates
in L2(7) is called a nonharmonic Fourier series. As noticed at the beginnin of the chapter, this is the context in which frames were originally define 1
For a given sequence A = {A&}fcGz, the frame radius is defined by
Я(Л) = sup{H : {elAkX}kez is a frame for L2(-R, H)}.
If {e1XkX}kEz is a frame for L2(—R, R) for some R > 0, it is automatically a frame for Z2(—R',R') for all Rf G]0, R] (Exercise 7.10). Writing
IR+ =]0,K(A)[ U {Я(Л)} и ]Ж), oo[,
we therefore have that
•	{ezXkX}kE% is a frame for L2(—R,R) whenever R e]0,l?(A)[.
•	{ezXkX}ke% is not a frame for L2(—R, R) if R G]H(A), oof.
The case R = R(A) itself is critical: there are cases where {elXkX}kez is a frame for L2(—H(A), H(A)), and cases where it is not.
A separated sequence {Afcjfcez is said to have uniform density d > 0 if
there exists a number L > 0 such that
k
Xk - з < L, \fk e Z.
a
(7.21)
Duffin and Schaeffer proved the following impressive theorem. We encourage the reader to consult the original paper [121] for the proof:
Theorem 7.6.1 Assume that {Xk}kez is a> separated sequence with uniform density d > 0. Then {A^J^ez has frame radius at least ird.
Theorem 7.6.1 can naturally be considered as a perturbation result. In fact, if we consider a fixed d > 0, then {ezkx^d}kEZ is a frame for L2(—R, R) for any R e]0,7T</] (Exercise 7.5); Theorem 7.6.1 now tells us that if {Xk}kez is separated and (7.21) is satisfied, then {elAfcI}fcez is a frame for L2(—R) for any R G]0,7td[. It is immediately clear that we can not expect {ezXkX}kE.z to be a frame for L2(—R, R) for R > nd; that it also might fail for R = 7rd is more subtle.
Our purpose is to find conditions on a sequence {A^J^ez and an interval I such that {elXkX}kez is a frame for L2(I). We begin with the Bessel condition.
Lemma 7.6.2 Let {Xk}kez be a real sequence. Then the following are equivalent:
(i)	{Xk}kez is relatively separated.
(ii)	{ezXkX}ke% is a Bessel sequence in L2(—7г,тг).
(iii)	{elXkX}kez is a Bessel sequence in L2(Z) for any bounded interval I C HL
7.6 Frames of exponentials
159
proof. A proof of (ii) о(iii) is outlined in Exercise 7.6. That (ii)=>(i) can be proved by an argument similar to the one used in Theorem 7.4.1; alternatively, it follows from Proposition 7.4.1; see Exercise 7.7. A proof of (i) => (iii) can be found in [279]. We also note that this implication actually follows from Theorem 7.6.1. In fact, it is enough to prove that {eiXkX}kez is a Bessel sequence if {Afcjfcez is separated. Assuming that {Xk}k£Z is separated we order {Xk}kez increasingly; we denote the reordered sequence by {Xk}keK- Depending on the given sequence {AjJ/cez, the index set К can be either Z,N, or = {—1, —2,... }. Now,
I A/c+i - Afc| > 6 > 0,
for some 5; therefore
A/c+i _ Afc 6	5 " '
By enlarging {Хк}кек if necessary, we can obtain a separated sequence {pk}kez> which, by choosing the ordering and indexing appropriately, satisfies that
|fc - у | < 1, Vk 6 Z.
By Theorem 7.6.1, the sequence {e^kX/5}kez is a frame for L2(I) when I is sufficiently small. Therefore {elfJ,kX}kez is a frame for L2(Z) when I is sufficiently small (Exercise 7.8); since {Xk}kez is a subsequence of {/ifcjfcez, this implies that {e1XkX}kez is a Bessel sequence in L2(Z).	□
We are now ready to prove a characterization of exponential frames due to Jaffard [179].
Theorem 7.6.3 Let {Xk}kez be a rea^ sequence. Then the following are equivalent:
(i) There exists an interval I such that {егХкХ}ке% is a frame for L2(Z).
(ii) {A^}fcez is the disjoint union of a sequence {A&J&eJi a uniform density di > 0 and a relatively separated sequence {X^kez^ 
If (ii) holds, then {elXkX}kez is a frame for L2(Z) for any interval I with |Z| < 2тгб?1.
Proof. Assume that {elXkX}kez is a frame for L2(Z) for some interval I. Then {Afcjkez is relatively separated by Lemma 7.6.2; by Lemma 7.1.3 this implies that for each integer N £ N we can find a finite number Cn such that each interval of the type [kN, (к 4- 1)7V[, к G Z, contains at most Cn elements from {A/J^z- By choosing N sufficiently large, we can assure that each interval [kN, (k 4- 1)АГ[, к 6 Z contains at least one element from {^k}kez', this is not trivial, but here is an argument. Assume the opposite,
160
7. Frames of Translates
i.e., that for each N 6 N we could find an interval [W, 4- l)/^, f G 2, which does not contain any element from {A^J^ez- Letting
fN(x) := e^+i^Nx,
it follows from Exercise 7.9 that
l(WAtI)l2 =
у ei((i+l/2)N-At)xrfa.
2	. (Xk-(£+l/2)N)
Afc-(£+l/2)№m\	2
4
|Afc - (£ + l/2)/V|2 '
Now consider an interval [n,n + l[, n e Z. If Afc e [n, n4-1[ for some к G Z, then the opposite triangle inequality shows that
|Afe - (£+ 1/2^| = |(n-(^4-l/2)^-(n-Afc)| > \n - (£ 4-1/2)ЛГ| - 1.
Using the above notation, at most Ci elements from {Xk}ke% belong to an interval [n,n 4- 1[. Also, if N > 4 and |n — (^ 4-	the interval
[n, n 4-1[ is contained in [£N, 4- 1)AT[ and therefore [n, n 4-1[ contains by assumption no element from {Xk}ke% *п this case. Putting all information together, we obtain that for N > 4,
Ei<wAtl>i2
k£Z
= E E l№,eiAtI)|2
n£Z {fc: Afce[n,n-+-1[}
-	JAfc - a 4- 1/2)ЛП
{n: |n—(£-+-l/2)JV|>^} №’• AfcG[n,n-+-l[}
_________4Ci__________
„ (|n-^4-l/2)7V|-l)2’
{n: |n-(£+l/2)N|>^} 4
So for N > 8,
El</w,eiAfcX>|2 fcez
4<?1	(|n| _ 2)2
{n: |n|>N/4} Vl 1	1
-> 0 as N -> oo.
Since ||/n|| = л/Й for all iV € N, it follows that the lower frame condition is violated. However, this contradicts our starting hypothesis that {егХкХ}ке^ Is a frame for L2(Z). This proves the claim that for N chosen sufficiently large, each interval [kN, (k 4- 1)АГ[, к G Z, contains at least one element from {Xk}kez-
Based on this we can now pick a subsequence of {Xk }kez having a uniform density. In fact, choosing N large enough we can for each interval of the form [2k N, (2/c-b 1)JV[, к e Z, pick one element from	belonging to
7.6 Frames of exponentials
161
the interval; this way we obtain a sequence {pk}kez = {^k}keii where the elements are separated by TV, and for which
\^k-2kN\ <N, Mel.
Finally we have to prove that the remaining sequence	\ {iak}kez
is relatively separated. One way to obtain a separated subsequence from {Afc}fcez\	is, for each к G Z, to pick one element from the sequence
belonging to each interval [2AW, (2 A; 4- 1)7V[ (if there is any); another separated subsequence is obtained by picking one element from each interval [(2k + l)N, (2k + 2)N[, A;GZ. After repeating these two procedures at most C/v times, no more elements from {A/Jfcez \ {p-k}kez are left. This proves that {A/J/cez \ {Цк}ке2 is relatively separated.
For the proof of (ii) =4* (i) we assume that there is a partition
Z = Ii U I2,
such that {Afcjfce/i has a uniform density di > 0 and {Xk}kei2 is relatively separated. By Theorem 7.6.1 the sequence {elXkX}keh is a frame for L2(—R, R) if R G]0,7rdi[; and by Lemma 7.6.2 {егХкХ}ке12 is a Bessel sequence. Therefore {eiXkX}kez is a frame for L2(—R, R) when R G]0, irdi [. By Exercise 7.10 this implies that {etXkX}kez is a frame for L2(Z) for any interval I with |Z| < 2тг<А-	□
The formulation in Theorem 7.6.3 is very convenient in order to prove that {elXkX}kez is a frame for a given sequence {Xk}ke%- In the original paper by Jaffard, Theorem 7.6.3 is just one step towards the main result, where the frame radius is determined for any sequence {Afc}&ez- Seip [254] gave a very elegant version of this final result in terms of the lower density:
Theorem 7.6.4 For {егХкХ}ке% to be a frame for L2( —7г,7г), it is necessary that {Afc}fcGz is relatively separated and D~({Xk}kez) > 1; and it is sufficient that {Afcjfcez is relatively separated and D~({Ak}kez) > 1-
Seip also proves that if {Xk}kez is separated and D~({Xk}kez) > 1, then {егХкХ}ке% contains a Riesz basis.
Theorem 7.6.4 is optimal in the sense that no conclusion is possible if ^“({A/Jfcez) = 1- For example, Seip proves that the sequence
{A,} = {fc(l - |/гГ1/2)}|к|>1
has density 1 and that {e1Afc:c} is a frame for L2(—7г,тг). On the other hand, a famous example by Kadec, namely the sequence {A/J^z given by
{A: — 1/4 if к > 0
к 4-1/4 if A; < 0 ,	(7.22)
0	if к = 0
162
7. Frames of Translates
also has density 1, however, without generating a frame for L2( —7г,тг). For a discussion of this example we refer to [279].
For a sequence {Д } in a general Hilbert space H, the upper and lower Riesz conditions in (3.24) are unrelated (Exercise 3.8). In the present context the situation is different: for a sequence {A/J^z consisting of distinct points, the existence of a lower Riesz bound for {elXkX}kez in L2(—7Г,тг) implies that {егХкХ}ке% is a Bessel sequence. That is, the lower condition is enough to guarantee that {егХкХ}ке2 is a Riesz basis for its closed span. This result was originally discovered by Young [280]. An elegant direct argument was later given by Lindner [211], and we repeat it here:
Theorem 7.6.5 Suppose that the sequence	consists of distinct
points and that there exists a constant A > 0 such that
for all finite scalar sequences {ck}kez- Then {elAfc:c}kez is a Riesz basis for its closed span in L2(—7г,тг).
Proof. Consider A^, Aj, where к j. By assumption, giAfcX _ eiXjX 112
2Л = Л(|1|2 4- |-1|2)

(7.24 >
Using the expansion
kl
k=0
it follows that for x e] - 7г,тг[,

|1 _ ei(Afc-Aj)z
kl
k—1
.|(Afc-A>|fc
kl
e]
Therefore (7.24) shows that 2A < 2тг(е^к M _ l)2; this implies that
Thus {Afc}fcez is separated, and therefore {elXkX}kez is a Bessel sequence in L2(—7г,7г) by Lemma 7.6.2.	□
7.7 Exercises
163
'The classical Kadec 1/4-Theorem states that if {Xk}kez is a real sequence for which supfcGZ < |,then {eiXkX}kez is a Riesz basis for L2(-tt, tt). The example (7.22) of Kadec shows that the conclusion fails if sup | A& — A; | = 1 Combining the proof of Kadec’s theorem in [279] with perturbation results for frames in Section 15.1 it is an easy matter to extend the result to frames; we refer to the original papers by Balan [12] and Christensen [67] for details.
Theorem 7.6.6 Let {A/J^z, {/ifc}/cez be real sequences. Suppose that {eifJ,kX}ke% is a frame for L2(—7v,7v) with bounds A,B. If there exists a constant L < 1/4 such that
frk — Afc| < L Vk e Z, and 1 - cos(ttL) 4- sinfnL) < у —,
then {elXkX}kez is a frame for L2(—7г,7г) with bounds
A(1 - y^(l “ cos(ttL) 4- szn(?rL)))2, B(2 — cos(ttL) 4- sinfr-L))2.
Compared with Kadec’ 1/4-theorem, the advantages of Theorem 7.6.6 are twofold: it applies to frames, and we obtain estimates for the frame bounds. Good values for the frame bounds are essential for estimates of the speed of convergence in algorithms involving frames, as we have seen already in Section 1.2. Recently, Theorem 7.6.6 has been used to construct Riesz bases for weighted L2-spaces consisting of solutions to certain Sturm-Liouville problems, cf. [167].
7.7 Exercises
7.1	Let ф 6 L2(R) \ {0} and let {Akjfcgz be a sequence in R. Show by a direct argument that {Тхкф}ке% can not be a frame sequence if {Afcjfcez has an accumulation point.
7.2	In this exercise we ask the reader to provide some details in the proof of Theorem 7.2.3.
(i)	Prove (7.10).
(ii)	Prove the equivalence between (7.11) and the statement preceding it.
164
7. Frames of Translates
7.3	This exercise connects to Exercises 3.8, 5.1. Let H be a separable Hilbert space.
(i)	Find a sequence {A}£i in H which satisfies the lower frame condition, but not the upper frame condition.
(ii)	Find a sequence {fk}kLi of vectors with norm 1, which satisfies the lower frame condition, but not the upper frame condition.
(iii)	Suppose that {elAfcX} satisfies the lower frame condition in L2(—7Г, 7г). Does it follow that {elAfcX} is a frame? Compare with Theorem 7.6.5.
7.4	Assume that ф has compact support and that there exist constants a, b > 0 such that a < |0(z)| < b for a.e. x 6 supp ф. Prove that {Тхкф}ке% is an orthogonal sequence for all sequences {Afc}fcez for which the sets {A^ 4- supp ф}ке2 are disjoint. How can the assumptions be modified to obtain a Riesz sequence which is not orthogonal?
7.5	Let d > 0, and prove that {elkx^d}kez is a frame for L2(—R, R) if and only if R e]0,7rd].
7.6	Let I and J be arbitrary bounded intervals in R and {Xk}ke% a real sequence. The purpose of this exercise is to prove that {егХкХ}кег is a Bessel sequence in L2(Z) if and only if it is a Bessel sequence in L2(<7); that is, the Bessel condition is independent of the considered finite interval. One way to proceed is to assume that {егХкХ}к^т is a Bessel sequence in L2(Z) and prove the following:
(i)	{elXkX}kez is a Bessel sequence in L2(a 4-1) for any a 6 1FL
(ii)	{elXkX}ke% is a Bessel sequence in L2(Zi) for any interval R С I.
(iii)	{егХкХ}ке% is a Bessel sequence in L2(Z U(a4-/)) for any a € B. Covering J with a finite number of translates of I we can now conclude that {elXkX}ke% is a Bessel sequence in L2(<7).
7.7	Let {Afcjfcez be a sequence in В and assume that {elAfcX}fcGz is a Bessel sequence in L2(—7г,тг). Define the function ф through ф — X[—, and prove that
(i)	{Е±±ф}ке% is a Bessel sequence in L2(B).
(ii)	{T_is a Bessel sequence in L2(]R).
(iii)	{A/J fcez is relatively separated.
7.7 Exercises 165
7.8	Consider an interval [5, с] C R, and identify L2(5, c) with a subspace of L2(R). For a given a > 0, let Da be the dilation operator. Prove the following:
(i)	DaL2(b,c) = L2(ab,ac).
(ii)	If {elXkX}kez is a frame for L2(6, c), then {егХкХ/а}ье% is a frame for L2(a6, ac).
7.9	Consider a bounded interval [5, с] C R. Let a 0 and prove that
7.10	Assume that {elXkX}kez is a frame for L2(I) for some interval I. Prove that {elXkX}kez is also a frame for L2(J) for any interval J with |J| < |Z|.
8
Gabor Frames in L2(R)
The mathematical theory for Gabor analysis in L2(JR) is based on two classes of operators on L2 (1R), namely
Translation by a e JR,	Ta : L2(JR) L2(JR), (Taf)(x) = f(x - a),
Modulation by b e JR,	Eb : L2(JR) -> L2(IR), (Ebf)(x) = e2lrlbxf(x).
Gabor analysis aims at representing functions f E L2(JR) as superpositions of translated and modulated versions of a fixed function g E L2(JR). There are two ways one can think about this. The first is to ask for integral representations involving all possible translations and modulations, i.e., representations like
f(x) = [°° Г Cf (a, b)e2~lb~g(x - a)dhda-	(8.1)
J — oo J — oo
here we have to search for a function Cf of two variables making this true. Note that we also have to specify in which sense we want (8.1) to be valid. The second approach is to restrict the translation and modulation parameters to a discrete subset Л C JR2 and ask for series representations of f in terms of the functions
{e^%(x-a)}w)eA.	(8.2)
The key to the first approach is the short-time Fourier transform, which we define in Section 8.1. Concerning the second approach, the natural question is how we can choose g € L2(JR) and the set A such that the functions in (8.2) constitute a frame for L2(JR). Formulated in this generality, the
168
8. Gabor Frames in L2(R)
question is very difficult, and we will mainly discuss the case where A is, lattice in R2; we actually saw the first example of such a frame in Example 3.7.2, where we proved that {EmTnXo,i]}m,nez is an orthonormal basis for L2(R).
The basic idea goes back to Gabor [143], who considered a sequence of functions of the form {ЕтьТпад}т,пе^ where ab = 1 and g is the Gaussian, g(x) = e~x2/2 (the same set of functions actually appeared already in the book [227] by Neumann in 1932 in the context of quantum mechanics). It was only observed much later (see the papers [181], [182] by Janssen and [114] by Davis and Heller) that this particular Gabor system leads to unstable expansions and is inappropriate for most applications. We come back to the exact meaning later, and just note that Davis and Heller proposed to overcome the difficulty by choosing a, b such that ab < 1.
The papers [181], [182] by Janssen can be seen as the starting point for the mathematical analysis of Gabor systems.
Gabor analysis took an entirely new direction with the fundamental paper [108] by Daubechies, Grossmann and Meyer from 1986. Here one finds for the first time the idea of combining Gabor analysis with frame theory. The authors constructed tight frames for L2(R) having the form {EmbTnag}m>nGZ at hand, and this contribution was the beginning of an intense activity which is still ongoing.
Parallel to this development, Feichtinger and Grochenig were studying expansions in Banach spaces in terms of coherent states (among which Gabor systems is a special case). In particular they obtained Gabor expansions in a large class of Banach spaces, which eventually lead Grochenig to introduce the concept of Banach frames. We will postpone a discussion of this subject to Chapter 17 and confine ourselves to Gabor analysis on L2(R) in this chapter and the next.
This chapter is the first among three, all dealing with Gabor frames. It contains the fundamentals, like equivalent conditions (and necessary, respectively sufficient conditions) for {EmbTnag}mine% being a frame. In order to provide a complete picture we also mention that Gabor systems are special cases of a larger class, called shift-invariant systems.
We begin in Section 8.1 by considering continuous representations. Then, after introducing Gabor frames in L2(R), we find necessary conditions for {ЕтьТпад}т,пе% to be a frame in Section 8.3. Sufficient conditions are given in Section 8.4.
Even when we restrict our attention to time-frequency shifts of the type {EmbTnag}mine%, it turns out to be very difficult to find the exact range of parameters a, b for which {EmbTnag}m>nEz is a frame for a given function g G L2(R). There are a few functions for which an exact answer is known; they are discussed in Section 8.6, together with the surprisingly difficult case where g is an indicator function of the type X[o,c], c > 0.
8.1 Continuous representations 169
The commutator relations (2.12) show that modulation in the time domain corresponds to translation in the Fourier domain. For this reason functions ЕьТад are called time-frequency shifts of g, and Gabor analysis is also known under the name time-frequency analysis.
The operators Еь and Ta will play a crucial role in this chapter. Note, that even though Еь is defined as an operator acting on L2 (JR), we will frequently use the same notation while the operator acts on another function space. For example, the symbol Eb alone will simply mean the function x е27ггЬх.
8.1 Continuous representations
Let us begin by motivating the definition of the short-time Fourier transform. For a signal /(ж), the variable x is often interpreted as time, and the Fourier transform /(7) gives information about the content of oscillations with frequency 7. In practice it is a problem that the time-information is lost in the Fourier transform, i.e., there is no information about which frequencies appear at which time. A way to try to overcome this problem is to “look at the signal at a small time-interval and take the Fourier transform here”. .This loose formulation means mathematically that we multiply the signal f with a window function g, which is constant on a small interval, and decays fast and smooth to zero outside the interval; by taking the Fourier transform of this product, we get an idea about the frequency content of f in the small time-interval. In order to obtain information about f on the entire time axis we repeat the process with translated versions of the window function.
This discussion leads to the definition of the short-time Fourier transform, also called the continuous Gabor transform:
Definition 8.1.1 Fix a function g e L2(IR) \ {0}. The short-time Fourier transform of a function f G L2(JR) with respect to the window function g is given by
Zoo _____________
/(x)s(x - y)e~2~'‘xydx, y,y e R.
-00
Note that in terms of the modulation operators and translation operators,
The short-time Fourier transform is the key to obtain a representation of the type (8.1):
170
8. Gabor Frames in L2(R)
Proposition 8.1.2 Let ft, f2, gi, g2 G L2(JR). Then
/СО poo	___________
/ ^gM(a,b^S2(f2)(a,b)dbda = (Л.ЛХл.й).
-oo J —co
Proof. By definition,
Ф91(Л)(а,Ь) = {fi,EbTagi)
POO	________
=	/ fr^e-^gtix-a^dx.
J —oo
Consider for a moment a fixed value for a. Then the above expression for’ ^i(/i)(a> &) is the Fourier transform of the function
Fi(z) = fi(x}gx(x - a),
evaluated at the point b. By introducing F2 similarly and using Plancherel’s and Fubini’s Theorems, we have
/ОО POO	___________
/	Ф91(/1)(а,Ь)Ф92(/2)(а,г>)</Ма
-	oo J —co
/ОО POO	_____
/ A(b)A(b)dMa
-	oo J —co
ZOO POO	_____
/ F^b)F2(b)dbda
-	oo J —oo
ZOO POO	____________
/ fi(b)gi(b-a)f2(b)g2(b-a)dbda
-	OO 7—00
ZOO	____ / POO ___________	\
/1(&)/г(6) ( / 9i(b - 0)32(6 - a) da] db
-	CO	\7—OO	/
= (A, /2) (#2,£1)-
□
Formulated directly in terms of the operators Еь,Та, Proposition 8.1.2 says that
ZOO POO
/ (fi,EbTagMEbTag2,f2)dbda = </г, /2X52,51)-
-co J —co
(8.3)
We now show how one can obtain integral representations like (8.1). Fix f 6 L2(JR); then Proposition 8.1.2 shows that the map
ZOO POO
/ UMW&g^dbda
-co 7 —oo
is a conjugated linear functional on L2(JR). By Riesz’ representation theorem there exists a unique element in L2(JR) — we call it
ZOO POO
/ (f,EbTagi)EbTag2dbda
-oo J —oo
8.2 Gabor frames
171
such that for all /2 € L2(IR),
[	[ {f,EbTagi}EbTag2dbda,f2
J —00 J —00
[ {f,EbTagl){EbTag2,f2)dbda
J —00 J —00 = (/,/2)^2, 91) •
These considerations lead to
Corollary 8.1.3 Choose ^1,^2 £ L2(IR) such that /0- Then every f E L2(IR) has the representation
f = Г Г {f,EbTagOEbTag2dbda,	(8.4)
{92,91) J- oo J—oo
where the integral is interpreted in the weak sense.
Thus we have obtained representations like (8.1) and explained how they have to be interpreted. Note that the function f 6 L2(JR) is represented as a superposition of time-frequency shifts of one function p2 € with coefficients given by the short-time Fourier transformation of possibly another function gi.
Connecting with the theory for continuous frames we have
Corollary 8.1.4 Let g e L2(JR) \ {0}. Then {EbTag}a,beR is a continuous frame for L2(IR) with respect to M = JR2 equipped with the Lebesgue measure.
The integral over ]R2 in (8.4) can be replaced by integrals over growing compact subsets of IR2, as done in e.g., [153]:
Lemma 8.1.5 Let	be a family of compact subsets of№? for which
Ki С K2 C •  • Kn C • • • and |J Kn = R2; n=l
let f E TL and define
fn ’•= 7----г [ (f,EbTagi')EbTag2dbda.
\92,9i) JKn
Then Ц / — /n|| -> 0 Q>s n —> 00.
8.2 Gabor frames
We are now ready to define the main subject for this Chapter.
172	8. Gabor Frames in L2(R)
Definition 8.2.1 A Gabor frame is a frame for L2(R) of the forrfi-{ЕтЬТпад}гп,пе%, where a,b > 0 and g G L2(R) is a fixed function.
Frames of this type are also called WeyI-Heisenberg frames. The function g is called the window function or the generator. Explicitly,
EmbTnag(x) = eMxg(x - na).
Note the convention, which is implicit in our definition: when speaking about a Gabor frame, it is understood that it is a frame for all of L2(R), i.e., we will not deal with frames for subspaces.
The Gabor system {EmbTnag}mine% only involves translates with parameters na, n G Z and modulations with parameters mb, m G Z. The points {(па,тЬ)}т,пе% form a lattice in B2, and for this reason one frequently calls {EmbTnag}mtnez a regular Gabor frame. In Section 9.7 we will consider more general sets of time-frequency shifts; in fact, we will let {(Дп,Лп)}п€/ be an arbitrary countable subset of B2 and investigate the frame properties for sets of functions of the type
{e^ix-xg(x - Мп)}пел	(8.5)
To distinguish between the cases, we will call a frame of the form (8.5) an irregular Gabor frame.
Ron and Shen have characterized all Gabor frames of the type {EmbTnag} We need a lemma before we state their result.
Lemma 8.2.2 Let fig G L2(R) and a,b > 0, к G Z be given. Then the series
f(x - na)g(x — na - k/b), x G R,	(8.6)
n£Z
converges absolutely for a.e. x G R; it defines a function with period a, whose restriction to [0,a] belongs to	In fact,
[ж У2 \ f(x ~~ na)g(x — na — A;/5)| ] G L1(0,a). n€Z
Proof. Since fiTk/bg e L2(R), we have fTk/bg G LX(R). Thus
/ У? I/(T ~ na)g(x - na - k/b)\dx = J' |/(т)^(т - A;/b)| die
< oo.
Therefore £neZ \f(x - nafjfa — na — k/b)\ < oo for a.e. x G [0,a]. Now when we know that the series in (8.6) converges for a.e. x G [0, a[, we can also conclude that it defines a function with period a, and the conclusion follows.	,
8.2 Gabor frames 173
Given g € L2(R), consider the matrix-valued function
M(x) := (9(ж - na - m/b))m neZ , x g R;	(8.7)
to be more exact, M(x) is for a.e. x G R well defined as a bi-infmite matrix, whose entry in the m-th row and n-th column is
Mm>n(x) = g(x - na - m/b).
Letting M(x)* denote the conjugated transpose of M(x), we formally consider the matrix product
M(x)M(x)*,
$yhose entry in the 772-th row and /с-th column is
С^Дя) = g(x — na — m/b)g(x — na — k/b).	(8.8)
nez
Note that the series defining Gm,k(x) is convergent for a.e. x G R by Lemma 8.2.2. The functions Gmik will also play a role in later sections.
When {cfcjfcez is a finite sequence, we can formally define the matrix product M(x)M(x)*{ck}kez, the outcome is a sequence, whose m-th entry is
У2 ~ na ~ mltyg(x - na — k/b)ck-nezkez
It turns out to be a necessary condition for {ЕтьТпад}т,пе% being a Gabor frame that M(x)M(x)* defines a bounded operator mapping /?2(Z) into £2(Z). Assuming that this is the case, we consider again a finite sequence {ck}ktz and obtain that
;	(М(х)М{хУ{ск},{ск})
= УУ У2 У? 9^x ~na~ m/tygix — na — к/Ь)скс^ ntlkEZmEl
2
= У2 ~na ~ k/b)ck >0.
nez fcez
Thus, M(x)M(x)* is a positive operator on £2(Z); in operator terms,
M(x)M(x)* > 0 on ^2(Z).
The characterization of Gabor frames given by Ron and Shen is as follows:
Theorem 8.2.3 Let A,B>0 and the Gabor system {ЕтьТпад}m,n^z be given. Then {EmbTnag}mine% is a frame for L2 (R) with bounds A, В if and only if
bAI < М{х)М(хУ < bBI a.e. x,	(8.9)
where I is the identity operator on £2(Z).
174	8. Gabor Frames in L2(R)
A natural way to use Theorem 8.2.3 is first to prove that the matrices M(x) given in (8.7) define bounded operators on ^2(Z), with a uniform bound on the norms. Then the operators M(x)M(z)* are bounded as well, and the upper condition in (8.9) holds. To prove the lower condition in (8.9) it is now enough to consider finite sequences.
The original proof of Theorem 8.2.3 is in [245]. A more accessible approach is given by Grochenig in [153]. We will later derive Theorem 8.2.3 as a special case of a characterization of shift-invariant systems, see page' 196.
In many cases it is convenient to assume that either the translation parameter or the modulation parameter in a Gabor frame is equal to 1. Given an arbitrary Gabor frame {EmbTnag}m,nez this can be obtained by a scaling of g, i.e., by replacing g with a function of the type
Dcg(x) =
Proposition 8.2.4 Let g € L2(JR) and a,b,c > 0 be given; assume that {ЕтьТпад}т,пЕ% is a Gabor frame. Then, with gc : = Dcg, the Gabor family {Emb/cTnacgc}m,nez is a frame with the same frame bounds as {EmbTnag} m,nEZ-
Proof. Operators of the type Dc are studied in Section 2.5, and they are unitary. By Lemma 5.3.3 it follows that {ЕсЕтьТпад}т,пе% is a frame with the same frame bounds as {EmbTnag} Using the commutator relations in Section 2.5,
DcEmbTna — Emb!CEcEna ~ Еть/cEnacE ci
and the proposition follows.	Й
8.3 Necessary conditions
We now move to the question about how to obtain Gabor frames {EmbTnag}m,nEz for L2(IR). One of the most fundamental results says that the product ab decides whether it is possible for {EmbTnag}m,nei be a frame for L2(]R):
Theorem 8.3.1 Let g E L2(JR) and a,b > 0 be given. Then the following holds:
(i) If ab > 1, then {EmbTnag}m^Ei is not a frame for L2(JR).
(ii) If {EmbTnag}m,nez is a frame, then
ab — 1 ф-г> {ЕтьТпад}is a Riesz basis.
8.3 Necessary conditions
175
Thus, it is only possible for {EmbTnag}m,nez to be a frame if ab < 1, and the frame is overcomplete if ab < 1. The proof of Theorem 8.3.1 will use some of the results developed in this chapter, so we have to delay the proof till page 214. We note that one can actually prove a stronger result than (i): when ab > 1, the family {ЕтьТпад}т,пе% can not even be complete in T2(< cf. [243].
The assumption ab < 1 is not enough for {EmbTnag}m,nez to be a frame, even if g 0 0. For example, if a e] 1/2,1[, the functions {EmTnaX[o,i]}m,nGZ are not complete in L2(JR) and can not form a frame.
The following proposition gives a necessary condition for {ЕтьТпад}т,пе2 to be a frame for L2(]R). It depends on the interplay between the function g and the translation parameter a, and is expressed in terms of the function £o,o defined in (8.8); since this function will be used often, we simply write
G(a;) = ^2 |^(t - na)|2.	(8.10)
n£Z
Proposition 8.3.2 Let g € L2(JR) and a,b> 0 be given, and assume that {EmbTnag}m,nez is a frame with bounds A, B. Then
ЬА < \g(x - na)|2 < bB, a.e.	(8.11)
nez
More precisely: if the upper bound in (8.11) is violated, then {ЕтьТпад}m>n^z is not a Bessel sequence; if the lower bound is violated, then {ЕтьТпад}т)ПЕ% does not satisfy the lower frame condition.
Proof. The proof is by contradiction. Assume that the upper condition in (8.11) is violated. Then there exists a measurable set Д C R. with positive measure such that G(x) = \g(x — na)|2 > bB on Д. We can assume that Д is contained in an interval of length By letting
До = {z e Д I G(x) > 1 + bB},
&k = {хе A | -r-~—+ bB < G(x) < 1 + 65}, fcGN, к 4- 1	к
we obtain a partition of Д into disjoint measurable sets. At least one of them, say, Д&/, has positive measure. Now consider the function f = %лк1, and note that ||/||2 = |Д^ |. For n e Z, the function f Tnag has support in Av; since Д^/ is contained in an interval of length 1/b and the functions {\/bE\nb}mgz constitute an orthonormal basis for L2(Z) for every interval I of length 1/b, we have
E \{f,EmbTnag)\2 = E \{fT^,Emb}\2 = -	|/(x)|2 |P(x-na)|2^.
mez	mez	-o°
176
8. Gabor Frames in L2(R)
Thus
£ |(/,Д»»ад|2 = Г l/WP |g(x-na)|2dx m.nEZ	n(EZ 00
= 1 [ G(x)dx bJbk,
= (B + RvW) M’-
But then В can not be an upper frame bound for {EmbTnag}minEz. A similar proof shows that if the lower condition in (8.11) is violated, then A can not be a lower frame bound for {Em{>Tna^}m>nez-	□
In particular, a function g generating a Gabor frame {EmbTnag}mtnEz is necessarily bounded. Note that Proposition 8.3.2 gives a relationship between the frame bounds and the lower and upper bounds for the function G in (8.10).
8.4 Sufficient conditions
Sufficient conditions for {EmbTnag}minE% to be a frame for L2(JR) have been known since 1988. The basic insight was provided by Daubechies [105]. A slight improvement was proved in [172]:
Theorem 8.4.1 Let g C L2(JR) and a,b> 0 be given. Suppose that
ЗА, В > 0 : A < \g(x — na)|2 < В for a.e. x E К (8.12) nez
and
E
У? TnagTna+ig nez
< A.
(8.13)
Then {EmbTnag}mtnez is a Gabor frame for L2(R).
We present a more general result in Theorem 8.4.4, so we do not prove Theorem 8.4.1. Both are based on an identity which we state in Lemma 8.4.3. This lemma uses another calculation, which we need repeatedly and therefore state first:
8.4 Sufficient conditions 177
Lemma 8.4.2 Let fig E L2(JR) and a,b > 0 be given. Given n E Z we consider the function Fn E L^O, 1/6) defined by
Fn(x) = f(x — k/b)g(x — na — k/b).
fcGZ
Then, for any m E Z,
fl/b {f,EmbTnag) = Fn(x)e^imbxdx.
Jo
In particular, the m-th Fourier coefficient for Fn with respect to the orthonormal basis {61/2e2’ri7nb:c}Tnez for L2(0,1/6) is
cm = 61/Z (/, ЕтьТпад).
Proof. We have already in Lemma 8.2.2 seen that the series defining Fn converges absolutely for a.e. x E JR. Now,
(f, EmbTnag) =	[ f(x)g{x - na)e~2™nbxdx
J —oo
, О/ь	______________
= Е/ f{x-kIV)9{x-na-k/b}e-^imbxdx
гУъ /	______________\
= / У? ~	~ na ~ V&) I e 2irvrnbxdx.
Ugz	/
We leave it to the reader to justify the manipulations.	□
Lemma 8.4.3 Suppose that f is a bounded measurable function with compact support and that the function G defined by (8.10) is bounded. Then
E \{f,EmbTnag)\2
m,n&£
1 f°°
= - /	|/(t)|2C7(i)&
О J — oo
1 Г 00 ______ ___________________________________
+	/ f(x)f(x-k/b)22ffi(x~na)9(x~na~k/b)dx.
& k^o -o°	nez
Proof. Let n E Z, and consider the ^-periodic function
F„(x) = E/(x-fc/b)p(^ — na — k/b).
fcez
We have already given a general argument for Fn being well defined pointwise a.e., but our present assumptions give more; in fact, for a given x E JR the compact support of f implies that f(x — k/b) can be non-zero only for finitely many /с-values. The number of /с-values for which f(x — k/b) 0 is uniformly bounded, i.e., there is a constant C such that at most C A:-values
178
8. Gabor Frames in Z2(R)
appear, independently of the chosen x. It follows that Fn is bounded, so Fn €	1/6) П L2(0,1/6); in fact, even
E \f(x - k/b}g{x -na- k/b)\ € L^O, 1/6) П £2(0,1/6).
A;GZ
By Lemma 8.4.2, for all m, n e Z,
fl/b (f,EmbTnag} = Fn(x)e~27rmbxdx. Jo
(8-14)
Since {\/6£^mb}mez is an orthonormal basis for L2(0,1/6), Parseval's theorem gives
1
- Уо \Fn(x)\2dx.
(8.15)
The assumption on f being a bounded measurable function with compact support will justify all interchanges of integration and summation in the final calculation. This follows from the observation that
/ОО _____ _____________ ____________
|/(t)/(x - fc/6)| ^2 |p(x - па)д(т
na — k/b)\dx < oo. (8.16)
The verification of (8.16) and the proof that this is exactly what we need; is left to the reader (Exercise 8.4). Now, via (8.14) and (8.15),
£ E \(f,EmbTnag)\2
ntZ mEl
1 ___ M/b
IE/ \Fn(x)\2 dx. ° n&Jo
Writing
|K(x)|2 = Fn(x)Fn(x) = E	- e/bMx -na- (/b)Fn(x),
tel
8.4 Sufficient conditions
179
we continue with
^^\(f,EmbTnagrf
7l(zZ mCZ
i rl/b _______________
= ±E/	E/(x-£/6)P(x-na-€/b)Fn(®)<fc
6nezJo t&
|	7*00 ___
= T E /	- «a)K(a;)da;
6neZ'/““
= |E [ f(x)s(.x - nd) E /(» - k/b)g(x -na- k/b)dx (8.17) nezJ~°°	fcez
= T [ l/(z)|2 E 15(ж _ na')\2<ix
|	7*00 _ ______________________________________________
+- y^ / f(z)f(z - k/b) У^ g(x — na)g(x — na — k/b)dx. □ b fe^o^-°°	nez
Note that the proof of Lemma 8.4.3 relies strongly on summing over all m €%'.we need the fact that {VbEmb}mez forms an orthonormal basis for L2(0, l/Ь). On the other hand, the proof did not use that we were summing over all n G Z, so the assumptions actually imply (see (8.17)) that for all index sets I C Z,
ЕЕКЛ-Ет(,Тпа5)|2
n€l mGZ
I	7*00 ___ ___________________
= - У^ /	f(x)9(x ~ na) У^/(z — k/b)g(x — na — k/b)dx.
Recall from Proposition 8.3.2 that the condition (8.12) is necessary for {EmbTnag}mtnez to be a frame. If it is satisfied, an estimate of the second term in the expression in Lemma 8.4.3 shows that {ЕтьТпад}т,пЕ% is actually a frame for all values of b for which
E
A#0
У? T^gT^^g n£Z
(8.18)
This is one way to prove Theorem 8.4.1. A more recent result can be found in [116], where it is proved that if (8.12) is satisfied and there exists a constant D < A such that
У2 У^ — na)9(x — na — k/b)| < D for a.e. x G K, (8.19) n(EZ
then {ЕтьТпад}m,nez is a frame for L2(K). This is not a generalization of Theorem 8.4.1 in a strict sense: there are cases where (8.19) is satisfied but
180
8. Gabor Frames in L2(R)
(8.13) is not, and vice versa. The main point is that other conditions (that are easy to check) for {EmbTnag}m,nez to be a frame can be derived from (8.19), cf. Theorem 2.4 in [116].
Below we prove a result that is more general than the above results.
Theorem 8.4.4 Let e L2(K), a, b > 0 and suppose that
В :=
I sup E
0 xe[o,a] к&г
g(x — na)g(x — na — k/b) n€Z
OO.
(8.20)
Then {EmbTnag}m>nez is a Bessel sequence with upper frame bound B. If also
(8.21)
1 inf
Ь iG[0,a]
^2 |5(x-na)|2^g(x-na)g(x-na-k/b)
n£Z	k^O n€Z
then {ЕтьТпад}тп,пе% is a frame for L2(K) with bounds A, B.
Proof. Consider a function f 6 L2(JR) which is continuous and has compact support. By Lemma 8.4.3,
E \(f,EmbTnag)\2	(8.22)
m,n£Z 1
= T l/(z)|2 E -na)\2dx	(8.23)
J~°° nez
+ |E [ f{x)f{x-k/b)^g(x~Tw)g(x-na-klb)dx. (8.24)
We want to estimate (8.24). For к G Z, let
Hk(x) := У Тпад(х)Тпа+к/ьд(х)i	(8.25)
n€Z
we observe that Hk is well defined a.e. by Lemma 8.2.2. Now,
Ei^/b^wi
к #0
Т-к/ъ	Tna9(x)Tna+k/ьд(х}
k^O	n(EZ
Tna — к/ЬЗ^^'^'-Г'па.д^Х^ k^O n&Z
8.4 Sufficient conditions 181
Replacing к with —k (which is allowed since we sum over all к 0 and complex conjugating all terms, we arrive at
- 52 52
k^O	k^O n€Z
= EE a 4- к j b 9 ( x ) a 9 ( )
nGZ
= Ei^wi-
k^O
So
Zoo ___ _____________ _______________________
f(x)f(x - k/b) У^^(ж ~ na)9(x — na — k/b)dx
'°°	nez
£ Г |/(x)| \Tk/bf{x)\ \Hk(x)\dx
^o7-00
E Г vWmi \Tk/bf(X)\^\H^\dx
k^oJ-°°
(*).
Using Cauchy-Schwarz’ inequality twice, first on the integral, and then on the sum over к 0,
(*) < E ( Г I/WIWWmT ( Г \Tk/bf(^\2\Hk(x)\dx)1/2 k^O ^7-°°	/	\«/—oo	J
/	\	1/2
< IE Г 1/(<|адн
V/oJ~°°	/
/ oo	\ 1/2
xiE Г°
y^o7-00	/
/ oo	\ 1/2
= I Г |/(x)|2£|Hfc(x)Mx
yj-oo	fc^0	J
/	\ 1/2
X| Г |/(»)|2Е|Т-кЛЯ*(х)1‘Ь:
\7-°° л/о	/
= Г 1№)|2Е|я,(х)Мх.
182
8. Gabor Frames in _L2(R)
Note that the expression

£ |Hfc(x)| = £ £rnoP(a:)Tne+fc/bP(x)
k^O	k^O
defines a periodic function with period a. By (8.22) and the condition (8.20) we now have
£ \(f,EmbTnag)\2
m,nEl
|2
nEZ
У2 \9(x ~ wa)|2 4- У2 У? 9(x ~ na)g(x - na — k/b) nEZ
dx
k=£0
= | [ l/O)|2 £ £ g(x - na)g(x - na - k/b) kEZ nEZ
< в н/п2.
Since this estimate holds on a dense subset of L2(R) it holds on L2(JR) by Lemma 3.2.6. This proves the first part. If also (8.21) is satisfied, we again consider a continuous function f with compact support and obtain that
£ \{f,EmbTnag)\2
m,nEZ
2
X
\g(x - na)|2 - У^ g(x — na)g(x ~^a~ k/b) dx nEZ	k^O
n£Z
By Lemma 5.1.7 the lower frame condition actually holds for all f G L2(R).
This completes the proof.	□
We have given a direct proof of Theorem 8.4.4, but it can also be proved as a special case of Theorem 8.2.3; see [153].
We also note that Theorem 8.4.4 can be extended to a result concerning frame sequences. We have seen that if the function
G(x) = £|5(x-na)|2	(8.26)
nEZ
is not bounded below, then {E^Tna^m^EZ can not be a frame for L2(JR). However, it can still be a frame for its closed linear span: in [45] it is
8.4 Sufficient conditions
183
proved that if the conditions in Theorem 8.4.4 hold with the infimum over x E [0,a]	(8.21) replaced with the infimum over Ng := {ж : G(x) 0},
then {EmbTnag}m,nEi is a frame for L2(Ng). This gives a way to construct multi-window Gabor frames: if gi,#2, • • • ,9k is a collection of functions which satisfy the conditions in this extended version of Theorem 8.4.4, then {EmbTnagk}m,n&,k=i,...,K is a frame for L2(U^Cj^J. In particular, if UjLi-Ngt = R, then we obtain a frame for L2(R).
The condition (8.20) is sometimes called condition (CC). It is not necessary for {ЕтьТпад}т,пЕ% to be a Bessel sequence, cf. [53]. In Section 9.1 we relate it to other conditions used in Gabor analysis.
Let us compare Theorem 8.4.1 and Theorem 8.4.4. Using the definitions of G and Hk in (8.26) and (8.25), we see that the conditions in Theorem 8.4.1 imply that
sup £ |ЯА(Ж)) <£ Halloo < inf G(x).	(8.27)
The advantage of Theorem 8.4.4 is that we compare the functions G and 1^(ж)1 pointwise rather than requiring that the supremum of |#fc(z)| is smaller than the infimum of G(x). For a given function g and a fixed value of a, this will usually imply that we can prove that {ЕтьТпад}т,пе% a frame for a larger range of the parameter b. The following example demonstrates this in practice and illustrates the different conditions graphically.
Example 8.4.5 Let a = b = 1 and define
1 4- x h 0
р(т) = <
if x e]o, 1], if x e]i,2], otherwise.
Consider for n, к 6 Z the function x -> g(x — n)g(x — n — k) for x G]0,1]. Due to the compact support of g it can only be non-zero if n 6 { — 1,0}; for n — —1 it can only be non-zero for к € {0,1}, and for n = 0, it can only be non-zero for к 6 {-1,0}. Therefore
g(x - n)g(x - n - fc) = n£Z
g(x)g(x 4-1)	if к = -1,
g(x)2 4- g(x 4-1)2 if к = 0,
<	g(x + l)g(x)	if A: = 1,
0	otherwise,
1(1 4- x)2 if к = —1, |(l + z)2 if к = 0, |(1 4- xj2 if к = 1, 0	otherwise.
So
G(x) - У2 - n)i2 = + -1)2’x £]°> n£Z
184
8. Gabor Frames in L2(R)
and
52 1я*(ж)1 = 12 529<^~n^(x~n~k) fc?^O	k=j£O nEZ
= (1 + x)2, X e]o, 1].
Theorem 8.4.4 now shows that {ЕтТпд}т,пЕ% is a frame for L2(K) with bounds A =	= 9. But inf^o^C^) = | and
E ii^iioo = 4,
k^O
so condition (8.18) is not satisfied. (8.19) is not satisfied either.
Figure 8.1 illustrates this. The inequality (8.27) can only be satisfied if there is a “horizontal gap” between the graphs of the functions G(x) and |Hk(x)|. This is clearly not the case if a = b = 1. But Theorem 8.4.4 applies if there is a positive minimal distance between the graph for G and the graph for |Hk(x)|, and this requirement is satisfied for a = b = 1.
Let us still consider a = 1 but now allow b to vary. For sufficiently small values of b all the discussed results will imply that {EmbTng}miTlEZ is a frame. Figure 8.2 shows that for b = 0.52 there is a horizontal gap between the graphs of the functions G(x) and ^2к^о 1^(ж)1» so Theorem 8.4.4 implies that {ЕтЬТпд}т1пе% is a frame. It is not clear from the graph whether (8.18) is satisfied, but at least the criterion (8.27) is satisfied.
Since {EmbTnag}minez can not be a frame if ab > 1, we know that even the criterion in Theorem 8.4.4 must break down for all b > 1. See Figure 8.3.	□
For a function with a sufficiently small support, the necessary condition (8.11) is enough for {EmbTnag}minez to be a frame. We also obtain very convenient expressions for the frame operator and its inverse in this case:
Corollary 8.4.6 Let a,b > 0. Suppose that g 6 L2(JR) has support in an interval of length | and that the function G satisfies (8.11) for some A, В >0. Then {EmbTnag}mine% is a frame for 722(JR) with bounds A,B. The frame operator and its inverse are given by
b	Cj
Proof. That {EmbTnag}minzz is a frame follows directly from Theorem 8.4.4 because
g(x - na)g(x — na — k/b) = 0 for all к 0 0. nEZ
8.4 Sufficient conditions 185
. The graphs of the functions G (the upper graph) and ^2k^Q |Hfe| for 1.
. The graphs of the functions G and l^fc| for a = 1,5 = 0.52. for G is the same as in Figure 8.1.
186
8. Gabor Frames in £2(R)
Figure 8.3. The graphs of the functions G and \Hk\ for a = 1, b = 1.1. The graph for G is the same as in Figure 8.1.
Given a continuous function f with compact support, Lemma 8.4.3 implies that
(Sf,f) =	£ \{f,EmbTnag)\2
т,п,ё£
1 f*00
= - /	\f(x)\2G(x)dx;
0 J —oo
by continuity of S this expression even holds for all f E L2(R). Via Lemma A.6.6 it follows that S acts by multiplication with the function y. □
For a continuous function g we can be even more explicit:
Corollary 8.4.7 Suppose that g E L2(R) is a continuous function with support on an interval I and that g(x) > 0 on the interior of I. Then {EmbTnag}m,na is a frame for all (a, b) G]0, |/|[x]0, щ [.
Proof. By Corollary 8.4.6 it is enough to prove that the function G is bounded above and below for the given values of a, b. For the upper bound, we observe that since g has support in an interval of length the function g(x—na) can at most be non-zero for	+1 values of n E Z, independently
of the choice of x E R. Thus
G(*)< (ф+1)|1<<-
8.5 The Wiener space W
187
For the lower bound, let J be the subinterval of I which has the same center as I and length a. Then, for any given x G R, we can find n G Z each that x — na G J; thus
G(z) > inf |P(j/)|2 > 0.	□
ye J
§.5 The Wiener space W
Given a positive number a, the Wiener space is defined by
W := < g : R -> C \g measurable and ||PX[fca,(fc+i)a[||oo > < oo. (8.28) I	kez	J
One can prove that W is a Banach space with respect to the norm
I Ы I w,a = 52 ||^[fca,(fc+l)a[||oo-fcez
The space W is independent of the choice of a, and different choices give equivalent norms; both statements follow from the fact that if 0 < a < b, then (Exercise 8.1)
5 > II^^[fc6,(k+i)6[||oo < 2 5 II^X[fea,(fe+i)q[lloo	(8.29)
fcez	kez
~	52 Huffed,(fc+l)6[l|oo-
'	' fcez
That g G W means that g is bounded and decays so fast that the “local maximum function” к ||<7X[fca,(fc+i)a]||oo belongs to ^X(Z). In Exercise 8.1 we ask the reader to prove that W C LX(R) П L2(K) . The condition for being in W is strong enough to exclude many of the pathological functions, which play a role for the understanding of functions in L2(JR) but are of little practical interest (like functions in L2(K) which do not decay to zero whenever |x| -> oo).
Lemma 8.5.1 Let g E W and a > 0 be given. Then
£|р(х-па)|<||р||^в> a.e. a: e R n£Z
If also heW and b G]0, |], then
52 52 9(x — na)h(x — na — k/b) ke% nez
< 2 ||p|| w,a||^||iv,a» a-e. x G JR.
188
8. Gabor Frames in L2(R)
Proof. For the first part, fix x G R, and observe that for .any given n eO there exists exactly one value of к G Z such that
x — na G [ka, (к + l)a[, к G Z;
furthermore, different values of n lead to different values for k. Therefore, £2 IpO - na)l < J2 llsX[ka,(fc+i)a[lloo = ||p||iv,a, a.e. X. nez	fcez
For the second part, we have
g(x — na)h(x — na — k/b) < У^ |p(x - na)| У^ \h(x — na - k/b)\. ke% nez	nez	kez
The first part of the lemma (applied to the function h and the translation parameter |) combined with (8.29) gives that
^\h(x-na-k/b)\<\\h\\W'i< 2 H/гЦил,o keZ
and the lemma follows.	□
If g belongs to the Wiener space, then {ЕтьТпад}тп,п^ is a Bessel sequence for all a, b > 0:
Proposition 8.5.2 If g G W and a,b > 0, then {EmbTnag}mtnEz is a Bessel sequence. If ab < 1, then В | ||р|1ила гз an upper frame bound.
Proof. The case ab < 1 follows immediately from Lemma 8.5.1 combined with Theorem 8.4.4. In case ab > 1, we can choose N G N such that ^b < 1; this implies that {ЕтъТп.^ g}m,nez is a Bessel sequence, and therefore the subsequence {ЕтьТпад}т,пе% is also a Bessel sequence.	□
An intuitive explanation of Proposition 8.5.2 is that functions in W decay relatively fast: given x G [0,a], the values of the functions
n, к н-> g(x — na)g(x — na — k/b)
are small enough to make	~ na)g(x — na — A;/5)| conver-
gent, with a bound independent of x. Actually, if we fix a > 0, the decay of g G W is enough to imply that if the necessary condition in Proposition 8.3.2 is satisfied, then {^m6^na^}m,nGZ is a frame for all sufficiently small values of b:
Proposition 8.5.3 Let g G W and a > 0 be given. Assume that there exists a constant C > 0 such that
С < У") |p(x - na)|2, a.e.
n€Z
8.5 The Wiener space W 189
{ЕтъТпа9}т,пЕ1 is o> frame for L2(JR) for all sufficiently small b > 0. b -> 0, the ratio between the frame bounds in Theorem 8././ converges
8иР£пёг1з(ж~па)12 ‘nfEn6z^U'-na)|2 ‘
proof. Proposition 8.5.2 shows that {ЕтьТпад}т,пег is a Bessel sequence for all b > 0. Fix e > 0 (the value will be decided later) and choose N G N such that E|„|>w llpXino.Cn+ijailloo < «• Letting g0 := gX[-aN,aN] and 9l:=g-go, we have
||Pl||w,a = 5 ' ||(g ~ aN,ajV])X[na,(n+l*)a[l|oo
nGZ
- 53 ll^[na.(n+1)a[lloo
|n|>N
<	6.
Now,
52 52 g(x ~ па)$(ж -na ~ к/ъ)
k^O nE'Z
=	^(.Яо + gi)(x-na)(g0 + gi)(x-na-k/b)
k^O n^Z
<	У^0(х - na)gp(x - na - k/b)
k^o nez
+ 52 52 So - na^ (x - na - k/b) k^O n£Z
+ 52 52 91 (x “ na)go(x - na - k/b) k^o nez
+52 5291 (x ~na^91 (x~na- к/ь) k^o nez
The function go has support in an interval of length 2aN, so if we choose b so small that | > 2aN, then the first of the above four terms is zero. Using Lemma 8.5.1 on the remaining terms, we get
52 52 9^x ~ na)9(x - na - k/b)
k^O n€%
< 4 | |po|| W,a\|#11| W,a + 2 ||P111pv,a
< 4c | |#o 11 w,a 4- 2e2
— 4c	4- 2c2.
190
8. Gabor Frames in L2(R)
If € is chosen such that 4б||£|| w,a + 2e2 < C, then the condition in Theorem 8.4.4 is satisfied, and {EmbTnag}is a frame.
The statement about the ratio of the frame bounds follows directly from the expression for the frame bounds in Theorem 8.4.4.	Qj
Proposition 8.5.3 is formulated as an existence result, and the usable values of b are somewhat hidden in the proof: after choosing e such that 4e||p||w>a 4- 2e2 < C, we have to choose N “large enough”, and then all 6 < 2^77 lea,d t0 a frame. However, for a concrete function g we will often be able to follow the proof directly and find an interval ]0, bo] explicitly such that {EmbTnag}rn,nez is a frame for all b G]0, bo] (Exercise 8.2). But the same exercise shows that it in practice might be preferable to work directly with Theorem 8.4.4 because the estimates used in the proof of Proposition 8.5.3 make bo unnecessarily small.
The second part of Proposition 8.5.3 is very relevant for applications: it shows that if g G W and SnGzl^(x — nn)|2 is “almost constant”, then g generates “almost tight” frames for small values of b. If we consider the function q(x) = TT-T from Exercise 8.2, then ' 7 l+x*	1
1-52 <	|S(x-n)|2 < 1.62, VxGR;
therefore, as b -> 0, the ratio between the frame bounds in Proposition 8.5.3 will be smaller than ~ 1.08.
8.6 Special functions
For a given function g 6 L2(B) it is usually difficult to find the exact range of parameters a, b > 0 for which {Em{,Tna^}m>nez is a frame. From Theorem 8.3.1 we know that a necessary condition is that ab < 1, and for a given value of a > 0 we can often use the proof of Proposition 8.5.3 to find an interval ]0, b0[ such that {EmbTna(;}m>nGz is a fr^ime for all b G]0, bQ[. However, Proposition 8.5.3 is based on Theorem 8,4-4, and the estimates used in the proofs make the results suboptimal. Also, the general characterization of Gabor frames in Theorem 8.2.3 is too difficult to use directly.
In this section we discuss some well-known functions and the range of parameters a, b for which they generate frames. First we consider the Gaussian:
Theorem 8.6.1 Let a,b > 0 and consider g(x) = e~x2. Then the Gabor system {ЕтьТпад}т'Пе% is a frame if and only if ab < 1.
The case ab = 1 is the easiest part. In fact, if {Em{,Tna^}m>nez was a frame for ab = 1, it would be a Riesz basis by Theorem 8.3.1; by the Balian-Low theorem this is clearly not the case. That the Gaussian generates
8.6 Special functions 191
a frame if ab < 1 was proved in 1991 by Lyubarski [216] and independently by Seip and Wallsten [254], [258] (that a fixed value of a > 0 will lead to a frame {Em{,Tna^}m>nez for sufficiently small values of b is clear from Proposition 8.5.3, but this is a much weaker statement). A historical note: Daubechies and Grossmann [107] proved around 1987 that {EmbTnag}is a frame whenever ab < 0.994 and conjectured the general result. The original proofs of the full result are complicated and use advanced complex analysis. Janssen gave later a shorter proof in [184]. A further analysis of the limiting case ab = 1 is given by Lyubarskii and Seip [217].
Another function for which the exact range of parameters generating a frame is known, is the hyperbolic secant, which is defined by
~ cosh(7rz) ’
This function was studied by Janssen and Strohmer [192], who proved that {EmbTnag}m,nez is a frame whenever ab < 1. The proof is based on Theorem 8.2.3 and the Zak-transform, which we introduce in Section 9.4. The hyperbolic secant does not generate a frame when ab = 1.
Let us now consider characteristic functions,
9 •= X[o,c[, c > 0.
The question is which values of c and parameters a, b > 0 will imply that {EmbTnag}m,nez is a frame. A scaling of a characteristic function is again a (multiple of) a characteristic function, so by Proposition 8.2.4 we can assume that b = 1. A detailed analysis performed by Janssen [190] shows that
(i)	{EmbTnag}m,nez is not a frame if c < a or a > 1.
(ii)	{EmbTnag}m,ne^ is a frame if 1 > c > a.
(iii)	{EmbTnag}min(zz is not a frame if a = 1 and c > 1.
Assuming now that a < 1, c > 1, we further have
(iv)	{EmbTnag}m,nez is a frame if a £ Q and c G]l, 2[.
(v)	{EmbTnag}minEz is not a frame if a = p/q G Q, gcd(p, q) = 1, and 2-4<c<2.
(vi)	{ErnETnagjrnjiez is not a frame if a > | and c = L - 1 4- L(1 - a) with L G N, L > 3.
'(vii) {EmbTna#}m>nez is a frame if |c - [cj - || < | - a.
The graphical illustration of this result is known as Janssen's tie. The surprisingly complicated structure for the characteristic functions indicates how complicated it is to find the exact range of parameters a, b which
192
8. Gabor Frames in L2(R)
generate a frame for a given function g. We return to a discussion of varying lattice parameters in Section 9.6.
A measurable set К С К is called a Gabor frame set if
is a frame for L2(K). It is absolutely nontrivial to classify such sets: Casazza and Kalton [58] have proved that this problem is equivalent to the longstanding problem of classifying the integer sets {ni < П2 < • • • < n^} for which the function f(z) =	does n0^ have any zero on the unit
circle in the complex plane.
8.7 General shift-invariant systems
In this chapter we have given a direct approach to Gabor systems, but this is just a choice: in fact, Gabor systems are special cases of what is known as shift-invariant systems, and several of the results considered so far are special cases of more general results. In this section we state the definition of shift-invariant systems and some of the important results with relevance for Gabor frames. The details and further results can be found in the paper [249] by Ron and Shen, and [185], [186] by Janssen.
Let {gm}m^i be a collection of functions in L2(K) and a > 0 be a given (shift) parameter. The shift-invariant system generated by {^m} and a is the collection of functions {gm(- — na)}mei,nez- Usually we will let I = Z, in which case we simply write
{*7nm}	{*7m(‘ "" nd) }m,n€Z-	(8.30)
Note the special case, where a shift-invariant system consists of translates of a single function; we studied this case in Section 7.2. Also, by taking
gm(x) = e2Mg(x), meZ	(8.31)
for a given function g 6 L2(K) and some b > 0, we obtain the shift-invariant system
{pnm} — {Tna£/mb^}m,nGZ-
Since
TnaEmbg(x) = e-^imnabe2Mxg(x - na),	(8.32)
the shift-invariant system {ТпаЕтЬд}т,пЕ% corresponds (up to an irrelevant complex factor) to the Gabor system {ЕтЬТпад}ш,пе%-
Frames and shift-invariant systems were first related in the paper [249] by Ron and Shen. Some of the most important results characterize frames of the form {gnTn} in terms of the functions gm or their Fourier transforms 9m- In particular, results of this type lead to equivalent conditions for a Gabor system {EmbTnay}mn(:7j to be a frame.
8.7 General shift-invariant systems 193
In this Section we follow the approach by Janssen in [185]. We will mainly prove results we need later, and refer to the original paper for other proofs and technical details. A key role is played by
Lemma 8.7.1 Assume that two shift-invariant systems {gnm} and {hnm} are Bessel sequences and let e,fE L2(JR). Then the function
p(e,/) : R —> C, p(e,/)(x) = £ {Txe^ Qnvri) {h nm i Tx f) m,nEl
is continuous and has period a. Its Fourier series in L2(0,a) is р{е,Ш = ^ске^а, kez
where
Ck = -	+ к/a) У2 9m(v}~hm(y + k/a)dv, keZ.
a J-°°	ma
Proof. The function p(e,/)(z) is well defined by the assumption that {gnm} and {hnm} are Bessel sequences (use Cauchy-Schwarz’ inequality); in fact, the series defining p(e, /)(z) converges absolutely for all x G R The continuity follows by a similar argument; in fact, given x,xq G R and letting В denote a common Bessel bound for {gnm} and {hnm},
|p(e,/)(x) - p(e,/)(z0)|
<	; |(Гге, Qn m ) {h nm, Txf) {TXQe,gnm}(h nm, TXof} I
m,nEZ
<	52 \(Txe ~ Tzoe,9nm}\ \(hnmi Tx f}\
m,nEZ
+ 52 I(Tzqe,gnm)I \{h птч Tx f-TXof}\ m,nEZ,
(\ 1/2 /	\ 1/2
52 \{Txe-Txoe,gnm}\2 j j	nm, Tx Л121
m,n£Z	j \m,nEl	j
(\ 1/2 /	\ 1/2
52 | (TXoe, ^nm)| 21	( E i<ftnm, Tx /-Tx0/)|2 I
77i,nEZ	j \тп,пЕ2	/
<	В ||rxe - TIoe|| ЦГх/ll +B IIT^ell ЦТ,/ - TXBf\\
=	В ||TMoe - e|| Ц/H + В ||e|| ||ТМ0/ - /Ц.
The last expression converges to zero for x -> xq by Lemma 2.5.1.
The periodicity of p(e, /) follows from the structure of the shift-invariant systems {д7ЛТП} and {hnrn}. For the computation of the Fourier coefficients we first assume that e, f are continuous functions with compact support;
194
8. Gabor Frames in L2(R)
this will justify all interchanges of sums and integrals. By (3.28), the coefficients in the Fourier expansion with respect to {e2infca:/a}fcGz are given by (Exercise 8.7)
Ck = - [a P(e,f }(x}er2^k^‘adx a JQ
= E E [a(Txe,gm(- - па))(Лт(- - na),Txf)e-2^adx
= -E Г <Txe,gm}{hm,Txf)e-2^adx
= -E	(8.33)
a meZ^-00
Now, for an arbitrary function ф G L2(JR),
{Тхе,ф} = {ЕТхе,Еф}
= {Е-Хё,ф}
=	/* e(i/)^Me"27ri:cl/di/
J — oo
= (e <£) (x).
It follows that
{Txf,hm)e2”ikx/“ = Я/а^(7Ц)(х)
— J~ ф'—k/a^f ^m)) (x).
Inserting the obtained expressions in (8.33) and using that the Fourier transform is unitary leads to
Ck = - E / (Txe,gm){Txf, hm)e2nikx/adx
° mfZ^30
=	- E Г T	^dx
amezJ~°°
=	- E Г
J —OO
meZ
I	r OO __________________л
=	“E /	+ k/a}hm{v + k/a)dv
amezj-x>
1 r°°	___________________
=	- eiy^fiy + k/aj^gm^hmiy + kla^dv.
a J-°°	mez
This proves the desired result in the case e,f G Cc(JR). The general case now follows by a density argument (Exercise 8.7).	□
8.7 General shift-invariant systems
195
We are now ready to state characterizations of several frame properties for shift-invariant systems. Given a shift-invariant system {gnm} as in (8.30), define the matrix-valued function
н(v) =	- k/a))k mez, a.e. v £ R.	(8.34)
Theorem 8.7.2 In the setting above the following holds:
(i) {#nm} is a Bessel sequence with upper bound В if and only if H(v) for a.e. и defines a bounded operator on £2(Z) of norm at most y/aB.
(ii) {gnm} is a frame for L2(K) with frame bounds A, В if and only if aAI < H(v)H(v)* < aBI, a.e. v.
(Hi) {$nm} is a tight frame for L2(K) if and only if there is a constant c > 0 such that
E 9mfr)9mfr + k/a) = c5k,o, к El, a.e. v. (8.35) mGZ
In case (8.35) is satisfied, the frame bound is A = с/a.
(iv) Two shift-invariant systems {gnm} and {hnm}, which form Bessel sequences, are dual frames if and only if
E+ k/a) = adk,o. к e Z, a.e. v. (8.36) mGZ
Proof. We only prove (iv) and derive (iii) as a consequence. For the rest we refer to [185]. The proof is based on the functions p(e, f) from Lemma 8.7.1 and the derived expression for their Fourier coefficients. Recall from Lemma 5.6.2 that Bessel sequences {gnm},{hnm} are dual frames if and only if
(e,/) = E (e, gnm) (h nmj Г), Ve,/6L2(R).	(8.37)
771,n£Z
If we assume that {gnm},{hnm} are dual frames, it follows from this identity that
p(e,f)(x) = (Txe,Txf) = Ve,fE L2(R).
Hence the function p(e, f)(x) and the constant (e, f) have the same Fourier coefficients in L2(0,a), i.e., for all к 6 Z,
1 roo ___________________________
-/	ё(г/)/(г/ + k/a)^ gm(v)hm(v + k/a)dv = 4,o(e,/)
J~°°	meZ
(i	= 4,о [ e(y)f(y)dv.
J —oo
196
8. Gabor Frames in L2(R)
Since this holds for all e 6 L2(K),
f(v + к/a) ^2	+ k/a) = a&.o/M, a.e. v, Vf € ba(R).
mgZ
For к = 0 this gives that
= a, a.e. v;
m£Z
furthermore the choices f = X[m/a,(m+i)/a[>m € lead
gm(v)hm(y 4- к/a) = 0, a.e. v when к 0. mgZ
The opposite implication is similar: assuming that (8.36) is satisfied, it follows that the functions p(e, f)(x) and the constant (e, f) have the same Fourier coefficients, so by continuity
p(e,/)(x) = (e,/), Vz e R
Now take x = 0, and we obtain (8.37).
The first part of (iii) follows immediately from (iv) combined with Lemma 5.7.1. For the second part we use the (unproved) result in (ii). If (8.35) is satisfied, the entry in the /с-th row and /-th column of H(v)H(v)* is
2*2	- k/a)gm(v - I/a) = c<5fcJ;
m6Z
by (ii), this implies that the frame bound is A = с/a.	□
Theorem 8.7.2 has direct consequences for Gabor systems (Exercises 8.8, 8.9). In Theorem 9.5.2 we will give a direct proof of one of them.
Theorem 8.7.2 characterizes frames of the type {gnm} in terms of the Fourier transforms gm. One speaks about characterizations in the frequency domain, as opposed to time domain characterizations directly in terms of the functions дш-
Theorem 8.2.3 is an example of a time domain characterization. We now show how it can be derived from Theorem 8.7.2 using that a Gabor system {EmbTnag}minez is a frame if and only if {77-1 Emt>Tnap}m,nGZ is a frame. Note that
•T7 EmbTnag = T—mbEnaJ’ g,
which is a shift-invariant system based on the translation parameter b and the functions EnaE~lg, n G Z. The k,n-th entry in the matrix H in (8.34) corresponding to this system is
FEnaF 1 gfy — k/b) = Tnag(v — k/b) = g(y — na — k/b).
That is, H equals the matrix M in (8.7), and the result follows.
8.7 General shift-invariant systems 197
The frame operator associated with a shift-invariant frame {gnm} commutes with the translation operators Tna\ in case of a single generator we proved this in Lemma 7.3.7, and it implies that the same holds for multiple generators. Therefore the canonical dual of {gnm} has the form
{S ^na9m} m/nE'Z, ~ {Тпа$ дтп}тп,пЕ^-
That is, for calculation of the dual frame it is enough to find the functions m G Z; the rest of the functions are obtained by translation. Calculation of S-1 gm can be done via a variation of Theorem 8.7.2 (iv). In fact, fixing у and letting {cm(i/)}mGz be the unique minimal-norm solution (see Theorem A.7.3) in ^2(Z) of the linear system
gm(y ~ k/a)cm(y) = abk,o, к G Z, a.e. y,	(8.38)
mCZ
it is proved by Janssen that
a.e. v.
For any dual frame {hnm}, Theorem 8.7.2(iv) shows that the sequence {Am(^)}mez solves (8.38); thus any dual frame {/inm} satisfies that
E	< £ |Am(i/)|2 , a.e. V,	(8.39)
mgZ	mgZ
with equality if and only if E(S~1gTn){y) = hm(y) for a.e. y. In the special case of a Gabor frame this yields an interesting result:
Proposition 8.7.3 Let g,h G L2(K) and a,b > 0 be given, and assume that {ЕтьТпад}тГ11'п£%, and {EmbEnah}т}п£$ wre dual frames. Then ||S~1g|| < ||< with equality if and only if S~rg = h.
Proof. Assume that {EmbTna#}m>nGz and {EmbTna/i}m>n€z are dual frames. Then also the shift-invariant systems generated by
gm(x) = e2™bxg(x), hm(x) = e^imbxh(x)
are dual frames. Thus (8.39) shows that
У2	“ mb)|2 <	|h(i/~m6)| , a.e. у, (8.40)
mgZ	mgZ
with equality if and only if
E(S~1g)(y — mb) = h(y — mb) V m G Z, a.e. у.	(8.41)
Now, integrating (8.40) over the interval [0,6] yields ||S-1^|| < ||h||, and (8.41) means exactly that S~rg = h.	□
Thus, there are two minimality properties associated to the canonical dual frame {E^T^S-1 ^}m>nGz:
198	8. Gabor Frames in L2(R)
•	The frame coefficients {(/, EmbTnaS~1g)}minEz minimize the A. norm of the coefficients in the frame expansion, cf. Lemma 5.4.2;
•	g minimizes the L2(JR)-norm among all functions generating aj dual frame with the Gabor structure.
Besides the references already mentioned, the reader can find more material on shift-invariant systems in e.g., [4], [32], [54].
8.8 Exercises
8.1	Let W denote the Wiener space.
(i)	Prove (8.29).
(ii)	Prove that W C L1^) A L2(JR).
(iii)	Prove that every bounded measurable function with compact support belongs to W, and that ||p||w,i < ( |supp(p)| 4- 1)
8.2	Consider the function g(x) = ттЦ-.
(i)	Show that g 6 W and find an estimate for ||p||w,i-
(ii)	Find a constant C such that
£2 \g(x - ")l2 >c, Vtsr. n£Z
(iii)	Show that for all N G N,
£2 IIPX[n,n+i]Hoo < 7Г - 2arctanGV - 1) +	2-
|n|>W	+	'
(iv)	Find via the proof of Proposition 8.5.3 a value bo > 0 such that {EmbTng}minez is a Gabor frame for all b 6]0,bo].
(v)	Estimate numerically via Theorem 8.4.4 the range of b for which {EmbTng}m,nEz is a Gabor frame.
8.3	Show by an example (maybe with a = b = 1) that the necessary condition in Proposition 8.3.2 does not suffice for {EmbTnag}minez being a Gabor frame. Similar statements with g replaced with g (and separate discussions of the lower respectively upper conditions) can be found in [20].
8.4	Prove (8.16) under the assumptions in Lemma 8.4.3 and justify ail the following interchanges of summation and integration in the proof.
8.8 Exercises 199
8.5	Prove that {EmTnaX[o,i]}m,nez is a frame for L2(R) for all a e]0,1].
8.6	Show that condition (CC) is satisfied for all a, b > 0 if g E W.
8.7	In this exercise we ask the reader to provide some details in the proof of Lemma 8.7.1.
(i)	Justify the calculations leading to (8.33).
(ii)	Provide the density argument at the end of the proof.
8.8	Derive a characterization for tight Gabor system {EmbTnag}m,n^ corresponding to Theorem 8.7.2 (for inspiration, see Theorem 9.5.2).
8.9	Prove via Theorem 8.7.2 that two Bessel sequences {ЕтЬТпад}тп}П^ and {EmbTnah}minez are dual frames if and only if
h(x - ma 4- k/b)g(x - ma) = b$k,o a.e. mez
8.10	Assume that {EmbTnag}minez is an overcomplete frame. Show that there exist dual frames not having the Gabor structure. (Hint: check the proof of Lemma 5.6.1.)
8.11	What is meant by the factor e-27rlmnab jn (8.32) being irrelevant?
Selected Topics on Gabor Frames
Beginning with the fundamental paper by Daubechies, Grossmann and Meyer [108] a very large number of articles about Gabor frames have been written. A complete description of Gabor frames would cover an entire book, and in this chapter we only discuss a few central themes. For more information about the role of Gabor frames in time-frequency analysis we refer to the book [153] by Grochenig. For a broader perspective on Gabor frames and their use in many different fields the reader should consult the two books [136], [137] edited by Feichtinger and Strohmer, which contain surveys and research articles covering several aspects.
We begin this chapter with a presentation of some conditions on a Gabor system {EmbTnag}minEz which appear repeatedly, and a discussion of the connections among them. Some of the conditions will be needed in Section 9.2 to assure convergence in re-writings of the frame operator.
In Section 9.3 we prove that the canonical dual of a Gabor frame again has Gabor structure. We discuss some of its properties and how other duals can be found. It is followed by a presentation of the Zak transform and its connections to Gabor systems in Section 9.4.
In Section 9.5 we characterize tight Gabor frames, while Section 9.6 analyses the role of the parameters a, b. Section 9.7 is devoted to Gabor frames where the lattice {(na, m6)}m>nGz is replaced by irregular sets in K2. We finally mention a few applications of Gabor frames in Section 9.8, and an alternative way of overcoming the Balian-Low theorem in Section 9.9.
202	9. Selected Topics on Gabor Frames
9.1 Popular Gabor conditions
There are some conditions on Gabor systems which appear in many different contexts. For later reference we collect them here and discuss their interrelations.
We already introduced the Wiener space W and proved in Proposition 8.5.2 that {EmbTnag} is a Bessel sequence for all a, b > 0 if g £ W. Another important subspace of L2(R) is known as the Feichtinger algebra 5o, introduced in 1980. Letting g(x) := e~x , it is defined as the vector space consisting of all f 6 L2(R) for which
[	[ \{ExTyf,g}\dxdy < oo.	(9.1)
J — OO J —oo
In terms of the short-time Fourier transform introduced in Definition 8.1.1, this means that Ф5(/) G L1(R2). So is a Banach space with respect to the norm
ll/ll^o = 11	(/) 11 l1 (R2) ’
and it is dense in L2(R). Several characterizations of So can be found in Feichtinger’s paper [124] and in [153]; in particular, So consists of all countable superpositions of time-frequency shifts of the Gaussian with F-coefficients:
s° = {/ = ^скЕУкТХкд : {(it,jk))cR2,{ct}ef2|.
The infimum of all -norms lcfc|> taken over coefficients representing a given f, gives an equivalent norm on So-
Nothing is special about g being a Gaussian in the definition of So', it can be replaced by any non-zero function in So, and (9.1) still characterizes all functions in So and leads to an equivalent norm. It can be proved that So C W, cf. [153].
For a Gabor system {EmbTnag}mtnEz, the set {(na, m6)}m>nGz C I2 is called the time-frequency lattice. It turns out that there are close relationships between frame properties for g with respect to this lattice, and frame properties with respect to the dual lattice, which is defined as the set {(n/Ь, m/a)}m>nGz. The first condition on a Gabor system we want to mention is related to this, and was introduced by Tolimieri and Orr [277] in 1995. A Gabor system {EmbTnag}minez is said to satisfy condition (A) if
£2 У^Ет/аТп/ьд}] <	И)
Condition (A) is often needed in order to guarantee certain convergence properties of infinite series appearing in Gabor analysis. However, as observed by Grochenig [153] it is preferable to avoid the condition if possible.
9.1 Popular Gabor conditions
203
for example, condition (A) is very sensitive to the choice of the lattice parameters: even for a simple function like g = X[o,i] and an arbitrary translation parameter a > 0, it is only satisfied for b = 1/q, q G N! Note that X[o,i] € W, i.e., stronger conditions are needed in order to avoid this kind of obstacle. One such condition is membership in Feichtinger’s algebra: in [153] it is proved that condition (A) is satisfied for all a, b > 0 if g e So-
Janssen introduced another condition in [189], which is frequently used in Gabor analysis. In contrast to condition (A), it only involves the function g and not the actual parameters a, b. We say that a function g G L2(R) satisfies condition (R) if
lim У^ - f \g(k + x) — g(k)\2 dx = 0.	(R)
еф0 fcez 6 \
2 6
Condition (R) might look restrictive, but it is actually satisfied for a dense class of functions in L2(K) (see Exercise 9.1 and page 243).
As we already proved in Theorem 8.4.4, a Gabor system {EmbTnag}min^z is a Bessel sequence if condition (CC) is satisfied, i.e., if
sup	g(x - na)g(x — na — k/b)
^e[o.a]A,eZ n6Z
oo.
(CC)
A variant of condition (CC) was used in [54]. We say that {EmbTnag}minEz satisfies condition (UCC) or the uniform condition (CC), if {ЕтьТпад}т,пе% is a Bessel sequence and for any given c > 0 there exists К G N such that
sup , E E g(x - na)g(x - na — k/b) z€[°,a] |fc|>K n£Z
< 6.
(UCC)
Note that a Bessel condition is part of the definition. Examples given in [54] prove that condition (UCC) is strictly stronger than condition (CC); this is no longer true if the Bessel assumption is removed (Exercise 9.2).
A slight modification of the proof of Proposition 8.5.2 shows that condition (CC) is satisfied for all a, b > 0 if g G W (Exercise 8.6). A detailed analysis of the relationship between the conditions is given in [54], where the following results are proved (the references are to the page numbers etc. in [54]):
•	if g G W, then condition (UCC) is satisfied for all a,b > 0 (p.110).
•	condition (CC) might be satisfied for functions g W (ex. 4.2).
•	If g G L2(1R) is positive and real-valued, then {EmTn<;}m>nGz is a Bessel sequence if and only if g satisfies condition (CC) (Cor. 3.7). The equivalence does not hold if the condition of g being positive is removed (Ex. 3.8).
204	9. Selected Topics on Gabor Frames
•	If {ЕтЬТпад}т, nEz is a Bessel sequence and g satisfies condition (A), then g satisfies condition (UCC) (Prop. 4.12).
•	There is a Gabor system {EmbTnag}minez satisfying condition (UCC) but not condition (A) (Ex. 4.13).
•	If ab G Q and {EmbTnag}m,nez is a frame satisfying condition (UCC), then also S~1g satisfies condition (UCC) (Th. 4.14).
9.2 Representations of the Gabor frame operator and duality
The structure of a Gabor frame turns out to have important implications for its frame operator, which can be rewritten in several ways. Many central frame results are based on the obtained representations of the frame operator. The analysis of the frame operator is very well covered in the literature: all results in this section are treated in Grochenig’s book [153], but we will also provide references to the original papers.
Walnut was the first to rewrite the frame operator S associated to a Gabor frame {ЕтЬТпад}т^п^г- In his thesis [284] from 1989 (see also [285]) he obtained what is now known as the Walnut representation: it expresses Sf in terms of the functions
Gk{x) = ^^g(x — na)g(x - na — k/b), к GZ.	(9.2)
n€Z
By Lemma 8.2.2 the series defining Gk(x) converges unconditionally for a.e. x G R In Walnut’s original work it was assumed that g belongs to the Wiener space W, but in [54] it is proved that it is enough to assume condition (CC).
Theorem 9.2.1 Assume that g G L2(JR) and a,b > 0 satisfy condition (CC). Then the frame operator associated to {EmbTnag}m,ne% has the representation
Sf=-b YSTb/bf)Gk.
kez
The series converges unconditionally in L2(R) for all f G L2(R).
If one only knows that {EmbTnag}mtnEz is a Bessel sequence and not that condition (CC) is satisfied, the Walnut representation is still valid for bounded compactly supported functions, but it might fail in general. Both results are proved in [54], which also contains a detailed analysis of the types of convergence that can occur if the representation does not converge unconditionally. Also, for an arbitrary Bessel sequence {ЕтЬТпад}тм^.
9.2 Representations of the Gabor frame operator and duality 205
large class of functions for which the Walnut representation nevertheless ^pnverges unconditionally is exhibited.
We already mentioned that there are close relationships between properties of a Gabor system {EmbTnag}m,nez and the Gabor system {Ern/aTn/b9}m,nez with respect to the dual lattice. Results of that type were obtained in a more general context by Rieffel [242]; for Gabor systems they were investigated almost at the same time by three groups of researchers, namely Daubechies, Landau, Landau [113]; Janssen [188]; and Ron, Shen [249]. There is a large overlap between their results, but their methods are quite different. A basic result is the following:
Lemma 9.2.2 Let g e L2(R) and a,b> 0 be given. Then {EmbTnag}minEz is a Bessel sequence with bound В if and only if {Em/aTn/bg}minE% is a Bessel sequence with bound Bab.
In the following results we will need the pre-frame operators associated to several Gabor systems with respect to different generators and different parameters. For this reason we need a more detailed notation than before, and we will denote the pre-frame operator for {ЕтЬТпад}тп,пе1 by Tg.aib instead of just T. With this notation, Lemma 9.2.2 states that Тд;а,ь is bounded if and only if	is bounded.
We now state a result from [113].
Proposition 9.2.3 Let f,g,h 6 L2(R) and a,b > 0 be given. If {EmbEnag}m,n€%>{EmbTnaf}min£% and {ЕтЬТпаЬ}т,пЕ% are Bessel sequences, then
^f-,o.,bTg.abh — ^ТМ/м/аТ;;1/ь>1/а/.	(9.3)
Proof. The complete proof in [113] is technical, and we will not provide all details. The main purpose of the following argument is to clarify how the dual lattice comes into play. We will prove Proposition 9.2.3 under the additional assumptions that f and h are compactly supported and bounded; this makes all needed interchanges of summations and integrals legal. First, let ф 6 L2(R). Then
Tf-,a,b<f> = {{Ф, Errtb'Bnaf') }m,nEZ-
By Lemma 8.4.2,
/•i/ь	__________________\
(Ф, EmbTnaf} = V ф(х - k/b)f(x -na- k/b) e~27rimbxdx. Jo \kez	J
206	9. Selected Topics on Gabor Frames
The interpretation of this equation in Lemma 8.4.2 in terms of Fourier coefficients together with Lemma 3.7.1 now gives that
(Ъ;а,ьт;;а^,ф) = (т^ьь,т^.ьФ)
—	(h, EmbTnag) (ф, EmbTnaf)
n^L m^Z
~ | E \ E “ Wet' -na-l/b),^ Ф(- ~ Vb)/(' -na- k/b)\ , nez \zez	fcez	/
where the inner product in the last line is in L2(0,1/6). When we write it out, we arrive at
1	z*1/6 /	_______________—
<Tf,a,bT^bh, Ф) = т^ E htx - 1ШХ -na- l/b} mgz
x ф(х - k/b)f(x — na — k/b) ] dx kez	J
2	r°°	_________________
= - У2 У2 /	~ 1/Ь)д(х ~na~ 1/Ъ)ф(х)ф(х ~ na)dx.
nez zez J-00
If we apply this calculation with other choices of the generators and th< parameters 1/6,1/a instead of a, 6, we obtain that
(^/i;l/b,l/a^;i/b,l/a/’ Ф)
/oo	____________________
h(x - m/b)g(x - ka - тп/Ь)ф(х) f(x — ka)dx.
‘°°
This shows that
(Е/-а,ьТд.аЬН,ф) = —{Th.i/b^/aTg.^ib^iaf^)\
since this holds for all ф 6 L2(K), the conclusion follows.	□
Written in terms of the involved sequences, (9.3) says that
(hi EmbTnag)EmbTnaf = — (f, Em/aTn/bg)Em/aTn/bh. (9.4) m,n£Z	m,nEZ
The right-hand side of (9.4) converges unconditionally in L2(R) because {Em/aTn/bh}7n,nez &nd {(/, Em/aTn/bg)}m,n£'z G £ (Z ) are Bessel sequences, see Lemma 9.2.2. We state some consequences of Proposition 9.2.3.
Corollary 9.2.4 Let g G L2(K) and a,b > 0 be given, and assume that {EmbTnag}minEz is a frame with frame operator S. Then the following holds:
9.2 Representations of the Gabor frame operator and duality
207
&(i) If h G L2(IR) and {EmbTnah}minez is a Bessel sequence, then
Sh =	(9,Em/aTn/bg)Em/aTn/bh.
m,nEl
(ii) s 9 = ^Ь^2т,пЕ^^ 19,Em/aTn/bS 19)Ern/aTn^bg.
Both follow from (9.4): for the proof of the first part, let f = g\ for the second part, replace h by g and replace g and f by S^g.
Janssen obtained similar results with slightly different assumptions in [188]. One result only assumes that {EmbTnag}minE% is a Bessel sequence, and delivers weak convergence of the frame operator for certain f 6 L2(IR); the second result requires that {EmbTnag}minez satisfies condition (A), and we obtain an unconditionally convergent representation:
Theorem 9.2.5 Assume that {EmbTnag}minez is a Bessel sequence with frame operator S. Then, for any f,h 6 L2(R) for which
E \{Em/aTn/bf,h}\2
m,nEZ
we have
{Sf,h) = — {9^m/aTn/bg)(Em/aTn^bf,h}\
the series converges unconditionally. If {ЕтЬТпад}т,пЕ% satisfies condition (A), then for all f 6 L2(JR),
=	{9,Em/aTn/b9}Em/aTn/bfi
m,nEZ
with unconditional convergence in L2(IR).
Condition (A) even implies that we have the representation
$	(9, Em/a^n/bg)-^тп/а^п/Ьч
I	m,nEZ
with absolute convergence of the series in operator norm.
Ron and Shen were the first to obtain some important results concerning the relationship between frame properties for a function g with respect to the lattice {(na,mb)}m^ni and with respect to the dual lattice {(n/6,m/a)}m>nz- Theorem 9.2.6 was published in [245]. Note that it, together with several other duality results, recently received a unified treatment by Grochenig [153].
Theorem 9.2.6 Let g 6 L2(IR) and a,b > 0 be given. Then the Gabor system {EmbTnag}m,nE% is a frame for L2(IR) with bounds A, В if and only if {Ei±T±g}mjl£% is a Riesz sequence with bounds abA,abB.
208
9. Selected Topics on Gabor Frames
The importance of Theorem 9.2.6 lies in the fact that it often is easier to prove that {E^T^g}m^z is a Riesz sequence than to prove directly that {EmbTna£}m,nGZ is a frame. We come back to an application of this idea in Section 16.4.
One often refer to Theorem 9.2.6 as the Ron-Shen Duality Principle. Another important duality result will be stated in Theorem 9.3.5.
9.3 The duals of a Gabor frame
It is important to notice that the frame operator S for a Gabor frame {EmbTnag}mynez commutes with the involved modulation and translation operators:
Lemma 9.3.1 Let g € L2(IR) and a,b > 0 be given, and assume that {ЕтьТПад}т,п£1 a Bessel sequence with frame operator S. Then
SEmbTna — EmbTnaS, Утп^п E Ta.
Proof. Let f € L2(IR). Using the commutator relations (2.8),
SEmbTnaf —	5	{EmbTnaf, EmibTn>ag}Em>bTn>ag
m' ,n'
-	52 (ft Т—паЕ(ш1 _m}bTnt ag}EmibTni ag
m' ,n' ez
=	E <Ле2™“(т'-т)ьЕ(т,_га)|,Т(^_п)ар)Ет^Т„.од.
m' ,n' GZ
Performing the change of variables m' —> m' 4- m, nr n' + n and using the commutator relations again,
S EmbTnaf
—	52 e I™™™ Ь {f i Em'bTn>ag)T\m,+m)bT(y+n}ag
m' ,n' GZ
=	E е“2”'гат'1’(/,-Ет'Л'а5)е2^’гат',,Ет(,ГпаЕт<ьГп(вд
m' ,n' gz
= EmbTnaSf.
□
As a consequence of Lemma 9.3.1, also 5-1 commutes with the operators EmbTna. Since S-1/2 is a limit of polynomials in S-1 in the strong operator topology, this operator also commutes with EmbTna. Thus, Lemma 9.3.1 has the following important consequence:
9.3 The duals of a Gabor frame 209
Theorem 9.3.2 Let g e L2(IR) and a,b > 0 be given, and assume that {EmbTnag}тп,пе% is a Gabor frame. Then the canonical dual also has the Gabor structure and is given by {ErnbTnaS~1 <?}m,nez- The canonical tight frame associated with {ЕтъТпад}m,n£Z is {EmbTnaS /2
Theorem 9.3.2 is very important for computations of the inverse frame associated to a Gabor frame: instead of calculating the double infinite family {S~lЕтпьТпад}^-^, it is enough to find S~}g and then apply the modulation and translation operators. It also gives a reason that even if {SnAa0}m,nGZ contains a Riesz basis as a subfamily, it might not be an advantage to remove elements from {ЕтьТпад}тп,пе%'- the computational benefits from the lattice structure of {(na, mb)}m>nGz will be lost, the operators EmbTna will in general no longer commute with the frame operator, and the dual frame will be more complicated to find.
The function S^g is often called the dual window function, and Theorem 9.3.2 is just one good reason to stick to the frame it generates when working with a Gabor frame. Bolcskei and Janssen showed in [39] that the canonical dual has other pleasant properties, which we now describe. It is based on a fundamental result by Jaffard [179], to which we will refer several times in the sequel:
Lemma 9.3.3 Suppose that {Ak,iw an invertible matrix and that there exist constants С, Л > 0 such that
|AM| <	VfcJ G N.
Then there exist constants C ,X' >0 such that
Vfc,/ G N.
The constants Cf, A' only depend on infцх||=1 ||Arr|| and вирц^ц-j ||Az||.
We say that a function g E L2(B) decays exponentially if there exist constants C, A > 0 such that
|<}(z)| <	a.e. x.
In [39], Lemma 9.3.3 is used to prove that if g decays exponentially and generates an overcomplete Gabor frame {EmbTna(;}mjnGz, then there exist constants C, A' such that
|S“ 1g(x)| < C'e~x a.e. x.
Again assuming that g generates an overcomplete frame, it is proved that exponential decay of g implies exponential decay of E(S~lg).
The same results hold with S~lg replaced by S~1^2g. In particular this leads to the following important statement about the canonical tight frame associated with {EmbTnag}m^a:
210
9. Selected Topics on Gabor Frames
Proposition 9.3.4 Let g G L2(]R)? and assume that g as well as g decays exponentially. Let a,b > 0,ab < 1 be given and assume that {EmbTnag}minEz is a frame; then S~l/2g as well as F(S~l/2)g decay exponentially.
Prior to [39], Bolcskei considered in [35] the case where {EmbTnag} mn^2 is rationally oversampled, i.e., ab = p/q for some p, q G N, q > p. He proved that if g is compactly supported, then S~xg is compactly supported if and only if the frame operator is a multiplication operator.
Despite the many nice properties of the canonical dual, we now discuss other duals associated to a given frame {EmbTna^}m,nGZ- There are several reasons to do this. First, one might be interested in duals minimizing other norms than the /?2-norm of the coefficients in the frame expansions. Second, there are cases where a function g generating a tight Gabor frame is badly localized; in this case one might prefer to search for a dual which is better localized than the canonical dual. Note that the question of finding other duals than the canonical is treated in the paper [113]: instead of searching for the dual minimizing the ^2-norm, the authors find, for an operator L on L2(]R) satisfying some conditions, a dual {EmbTna/i}m)nGz, for which \\Lh\\ < \\Lf\\ for all duals {Em&Tna/}m)nGZ.
The general characterization of all dual frames in Theorem 5.6.5 of course also applies to Gabor frames, but if {EmbTnag}mynE% is an overcomplete frame, not all of these duals have the Gabor structure (Exercise 8.10). The duals with Gabor structure are characterized in the famous Wexler-Raz Theorem [287]. Several proofs of this result exist: we will derive it as a consequence of Theorem 8.7.2.
Theorem 9.3.5 Let g,h G L2(R) and a,b > 0 be given. Then, if the two Gabor systems [EmbTnag^<nri,ri.£Z and \^E<mbTnQth^'m^n^'^J are Bessel sequences, they are dual frames if and only if
{h, Em/aTn/bg) = 0 for all (m, n) / (0,0) and {h, g) = ab. (9.5)
Proof. The Bessel sequences {EmbTnag}m,nez and {EmbTna^}m,nGZ are dual frames if and only if the shift-invariant systems {TnaEmbg}minEz and {TnaEmbh}m>ne% are dual frames. The generators for the two latter systems are gm = Embg and hm = Embh; by Theorem 8.7.2 they generate dual frames if and only if
§m(^)^m(^ + k/a) = a5kio, fc G Z, a.e. v.
TTlgZ
In terms of the functions g and h this is equivalent to
У2 9^ ~ mb)h(v 4- k/a — mb) = а^)0, к G Z, a.e. 1/.	(9.6
m£Z	\
9.3 The duals of a Gabor frame
211
We can express this condition in terms of the coefficients in the Fourier expansion with respect to {e'27rmi7/b}nGz for the 6-periodic functions
фк(Е) := g(v — mb)h(is 4- k/a — mb), к E Z : m£Z
in fact, (9.6) is equivalent to all coefficients for фк, к 0, being zero and the coefficients for ф0 being zero for к 0 and equal to a for к — 0. Wexler-Raz’ theorem is now a consequence of the following computation, which yields the n-th coefficient for the function фк in the Fourier expansion with respect to {e2’rinP'"’}neZ:
1 [\we-2*iw/bdv
Ь Jo
= y- f g(p — тЬ)А(г/ + k/a — mb)e~2,rm‘'^bdi' bJ° m&
1 roo ____
=	-	g^')h^ + k/a)e-2jr,ni,/bdiy
0 J —oo
—	^{T-k/ah-) En/bO)
=	| (Fh, FEk/aT_njbg)
= ~{h,Ek/aT_n/bg).
□
In the terminology used in Section 6.5, the Wexler-Raz theorem characterizes the functions h generating a dual Gabor system of a frame {EmbTnag} as the solutions to a moment problem with respect to the sequence {Em/aTn//b^}m)nGz. We can now use the general results for moment problems to find an alternative description of the generator S~Yg for the canonical dual frame. In the literature it is known as “the Wexler-Raz dual equals the canonical frame dual”.
Proposition 9.3.6 Let {ЕтЬТпад}т>пе% be a frame with frame operator S. Then S~Tg is the unique minimal-norm solution to the moment problem
(h, Em/aTn/bg) — o^n^ab.	(9.7)
Letting S denote the frame operator for {Em/aTn/bg}mtnEz, we further have
S~rg = abS~1g.
Proof. By Theorem 9.2.6 we know that {Em/aTn/bg}m,ne% is a Riesz sequence, i.e., a Riesz basis for H у^{Ет/аТп/Ьд}т,пЕ%. By Theorem
212	9. Selected Topics on Gabor Frames
6.5.1 and Exercise 6.4, the moment problem (9.7) has a unique solution belonging to H. This solution is in fact, S~1g is a solution by Theorem 9.3.5, and S~1g € H by Corollary 9.2.4. On the other hand, letting S denote the frame operator for {Em//aTn//b^}m>nGz, Theorem 6.5.2 shows that
S~1g = ab ^2 ^т,о^п,о8^Ет/аТп/ъд = abS~1g.	(9.8)
771,nGZ
All other solutions to (9.7) are obtained by adding an element f e H1- to the solution in H. Thus, the special choice (9.8) minimizes the norm among all solutions to (9.7).	□
We can also express the equations in (9.5) via an operator equation. Let
H : L2(R) -4 £2(Z2), Я/ = {(/, Em/aTn/bg}}m^z. (9.9)
Note that H is the adjoint of the pre-frame operator associated with the Gabor system {Em/aTn//b^}m)nGz. In terms of Я, (9.5) is equivalent to
Hh — Q-5{^77i,O^n,o}771.,77GZ-	(9.10)
Corollary 9.3.7 Let g € L2(IR) and a,b > 0 be given, and assume that {EmbTnag}77i,tiez a frame. Then
S~1g = аЬН^НН^ГЧ^п^т^	(9.11)
Proof. We again use that {Em/aTn/bg}minez is a Riesz sequence; it implies by Theorem 6.5.1 that the operator H in (9.9) is surjective. Thus, we know from Theorem A.7.3 that the minimal-norm solution to (9.10) can be expressed via the pseudo-inverse of H. Using (A.12), we obtain (9.11), as desired.	□
Equation (9.11) is known as the Janssen representation of the function generating the canonical dual of {EmbTnag}m,nez-
We can obtain a more concrete expression for S~1g. First we note that for any sequence {cm>n}m>nGz € £2(Z2),
HH {cm>n}m>nG2 — <
Em> !aTn> /Ъ9•> Ещ/а^п/Ьд
>
> 77l,72£Z
Let	be the canonical basis for /?2(Z2); that is, em>n is the
sequence in ^2(Z2) given by
^771,72 — {0772,771/^72,71'J 772',71'GZ-
We now re-index {em)n}m,nGZ as {e/JgTj in an arbitrary way such that ei corresponds to eo,o (Exercise 9.8); denote the corresponding re-indexing of {Em/aTn /b9}m,nGZ by {g^	. We can then represent HH* via its matrix
9.3 The duals of a Gabor frame 213
with respect to	, i.e., the bi-infinite matrix whose j/c-th entry is
and (9.11) takes the form (see Corollary 6.5.3)
oo
=	(9-12)
J=1
If €.j — em,n and =	then
{HH ek,e.j) = {Emi/aTni ^9, Em^aTn^g),
the Gram matrix for {Em/aTn/b9)}m,nez- We write for short
(H H }тп,п,тп' ,n' — {Em>/aTn./b9, -^тп/а^п/ЬЭ} , Ш,П,Ш , П Е (9.13) with this notation,
S^g = ab £ [(ЯЯТ’]™,»,о,о Em/aTn/bg. (9.14) 771,nGZ
For practical purposes it is not only important to characterize the duals of a frame {^mb^na^m.nGz: we also need to know how to find them. A constructive procedure to find some duals having the Gabor structure was given by Li [207]:
Proposition 9.3.8 Let g € L2(IR) and a,b> 0 be given, and assume that {EmbTnag}ni,nez is a frame for L2(R). Then, for any f £ L2(R) for which {EmbTnaf}m,nez is a Bessel sequence, the function
L	h —S1g-[-f — (S 1 g, ErnbT,na9)^mbTnaf
f	m,n€Z
generates a dual frame {EmbTnah}m,nez of {EmbTnag^m,nEZ-
Proof. Applying Lemma 5.6.5 we see that if {EmbTnaf}minEz is a Bessel sequence, then {EmbTnag}minez has the dual {/cm>n}m>nGZ given by
km,n — S EmbTnag d- EmbTnaf
.	~ {S EmbTnag, Em>bTn>ag}Em'bTn>af
m' ,n' ez
— EmbTna(S 9 + f)
~ {EmbTnaS 9,Em>bTn'ag)Em>bTn>af• m' ,n' ez
Exactly as in the proof of Lemma 9.3.1 one shows that
(EmbTnaS 19i EmibTn>ag)EmibTn'af m' ,n' ez
— EmbTna (S 1 g, ErnibTniag}ErnibTniaf\l
m' ,n' ez
И
214
9. Selected Topics on Gabor Frames
thus
— EmbTna I S 1 g + f — (S g, Em> bTn> ag) Em> bTn> a f j . C \	m,',n'EZ	j
One can say that the functions f € L2(JR) generating Bessel sequences {EmbTnaf}m,nE% give a parametrization of a class of dual frames of {ЕтьТпа^}Ш)Пег maintaining the Gabor structure.
We end this section by the announced short proof of Theorem 8.3.1:
Proof of Theorem 8.3.1: Assume that {EmbTnag}minez is a Gabor frame for L2(]R). We begin with an observation concerning the canonical tight frame {EmbTnaS~1/2 g}mynez associated with {EmbTnag}m,nEZ^ First, we apply Lemma 8.4.3 on the frame {ЕтЬТпаЗ~1^2д}т^пЕг- For an arbitrary bounded and measurable function f with support in an interval of length l/Ь we obtain that
Zoo
|/(i)|2dT = У |(/,EmbTnoS-,/2p)|2
°°	m,neZ
1 r°°
= r \f(x)\2 ^\s^1/2g(x-na)\2dx-, 0J-°° nSz
from here, a standard argument gives that J2nGZ 15“1//2p(x — na) |2 = b for a.e. x e R Therefore
Zoo
|S"1/2p(a:)|2da;
-oo
= [ ^\S~1/2g(x — na)\2dx nez
= ab.
For the first part of Theorem 8.3.1 we have to prove that ab < 1 for the arbitrary given frame {EmbTnag}m,nez- Now, since {EmbTnaS~l/2 д}туПЕг is a tight frame with frame bounds equal to 1, Exercise 3.5 implies that ||S-1//2g|| < 1. Combining with ||S-1//2^||2 = ab we obtain that ab < 1 as desired.
For the second part we have to prove that a frame {EmbTnag}is a Riesz basis if and only if ab = 1. First, assume that {ЕтЬТпад}туПе% is a Riesz basis. Then {EmbTnaS-1//2^}m)nez is a Riesz basis having bounds A = В = 1; in particular this implies that ||S-1//2^|| = 1. Since we have already given a completely general proof for the equality ||‘S'-1//2g||2 = аб. we have 1 = ab as desired.
For the other implication we now assume that ab = 1. Then
=ab= 1,
9.4 The Zak transform
215
and therefore |\EmbTnaS~ 1//2<?|| = 1 for all m,n e Z. Using Exercise 3.5 we conclude that {EmbTnaS~1^2 g}m,nez is an orthonormal basis for H, and therefore
{-E'mb^na3'}m,nGZ — EmbTnaS ^}m,nEZ
is a Riesz basis.	□
9.4 The Zak transform
The Zak transform is a very useful tool to analyze Gabor systems {EmbTna9}m,nez in the case where ab 6 Q. For a classical survey on the Zak transform we refer to the article [183] by Janssen; recent applications to Gabor analysis appear in e.g., [153], [39], [20].
For a fixed parameter Л > 0 we define formally the Zak transform Z\f of f € L2(R) as a function of two real variables:
(ZA/)(^O = A1/2^/(A(i-fc))e2’ri*:‘')	(9.15)
fcez
In the case Л — 1 we simply write
(Z/Жр) = ]T/(<-fc)e2’rifct', t.i/еж.	(9.16)
fcez
For functions f E Cc(JR) the Zak transform is defined pointwise, but for general functions in L2(IR) we have to be more precise about how to interpret the definition. Letting Q := [0, l[x[0,1[ we now prove that the series defining Zxf in fact converges in L2(Q) for all f E L2(JR):
Lemma 9.4.1 Given Л > 0, the Zak transform Z\ is a unitary map of L2(JR) onto L2(Q).
Proof. We first consider the case Л = 1. Let f € L2(JR) be given. In order to show that Zf is well defined as a function in L2(Q), we consider the functions
Fk(t,v) — f(t - k)e2^k\ к eZ.
These functions belong to L2(Q). Denoting their norm by |\Fk ||l2(qj, we observe that
En^iib(Q) = e/1 kez	kezJ° Jo
= E mt - k)\2dt
= ll/ll2-
216
9. Selected Topics on Gabor Frames
Furthermore, for j к,
{Fk, Fj}L2(Q) = f f{t - kjW^F) ( Г e2^k-^dJ\ dt = 0. (9.17) JO	WO	/
Combining the obtained results shows that Fk in fact converges in L2(Q) and that
2
£Fk =El|Ffc||b(Q) = Н/П2;
L2(Q)
thus Z is an isometry from L2(R) into L2(Q).
For the rest of the proof we use the Gabor basis {EmTnX[o,i]}m,nGZ for L2(R), cf. Example 3.7.2. By direct computation for (t,p) € Q,
(ZErornX[o,i])(t^) =
fcez
= e2’rlmte-2,rin1' ^2X[o,i](t - fc)e2’rikl' kez
= (9.18)
That is, the Zak transform maps the orthonormal basis {EmTnX[o,i]}m,nez for L2(R) onto the orthonormal basis {e-27rinMe27rimt}m)nez for L2(Q). This implies that Z is unitary.
For the general case we note that in terms of the unitary dilation operator Dx-i defined in Section 2.5,
Zxf = Z{Dx-i f).
As a composition of unitary operators, Zx is itself unitary.	□
Now where the Zak transform is proved to be well defined a.e. on Q, an inspection of the expression (9.15) reveals that Zxf(t,i/) is even defined a.e. on R2 and that the quasi-periodicity in Lemma 9.4.2(i) below holds. We collect some more properties of the Zak transform:
Lemma 9.4.2 Consider the Zak transform Zx, Л > 0, and f 6 L2(R). Then the following holds:
(i)	Zxf(t + l,y) = e2™Zxf(t,u), Zxf(l,u + l) = Zxf(t,u).
(ii)	If f is continuous and, for some C > 0,
then Zxf is continuous on R2.
(iii)	If Zxf is continuous on R2, then there exists G R2 such that Zxf(t,v) = 0.
9.4 The Zak transform 217
The proof of (ii) is similar to the proof of Proposition 7.3.6 (Exercise 9.3). (iii) is proved by Janssen in [182] and in [172]. Note that the quasi-periodicity in (i) often leads to jump-discontinuities on the lines t = k,k E Z : even if Z\f is continuous on Q it might not be continuous on B2 • For a concrete example, take the function f whose Zak transform is equal to 1 on Q: in this case Z\f is continuous on Q but not on B2.
If g E L2(M) and ab = 1, a computation as in (9.18) shows that
ZaEmbTnag = eMe-2*™Zag.	(9.19)
The family {e27rimfe-27rmi"}m)nGz is an orthonormal basis for L2(Q), which we denote by {E(wn)}m>nGz- The equation (9.19) shows that {ЕтьТПа9}тп,пЕ1 is complete in L2(R) (respectively, an orthonormal basis for L2(R) or a Riesz basis) if and only if {E(m n)Za^}m>nGz has the same property in L2(Q). This observation will be used in the following theorem, which expresses properties for a Gabor system {EmbTna</}m>nGz with ab = 1 in terms of the Zak transform Zag. Remember from Theorem 8.3.1 that a Gabor system with ab — 1 is a frame if and only if it is a Riesz basis.
Proposition 9.4.3 Let g E L2(B) and a,b > 0 with ab — 1 be given. Then the following holds:
(i)	{ЕтьТпад}т>пе2 is complete in L2(R) if and only if Zag 0, a.e.
(ii)	{ЕтьТпад}т,пЕ% is a Bessel sequence with bound В if and only if < B, a.e.
(iii)	{EmbTnag}m,nEz is a Riesz basis for L2(K) with bounds A,B if and only if A < \Zagl2 < B, a.e.
(iv)	{ЕтьТпад}т>пЕ% is an orthonormal basis for L2(R) if and only if \Zag\2 = 1, a.e.
Proof. To prove (i), consider the subspace V C L2(R) given by
V — {/ E L2(R) : Zaf is bounded}.
The bounded functions are dense in L2(Q), so V is dense in L2(R) by Lemma 9.4.1. Now let f E V. Then
(/, EmbTnag)L2^ = (Zaf,E{mtn}Zag)L2{Q}
— {ZafZag,E{mfn))L2{Q}. (9.20)
First assume that Zag 0 a.e.. If f 0, then ZafZag is not the zerofunction, and there exists (тп,п) E Z2 such that
(Zaf Zag, E
(m, n)>L2(Q) / 0-
Therefore (9.20) shows that {ЕтьТпад}т>пЕ% is complete. For the other implication, assume that Zag = 0 on a measurable set Д C Q with positive
218
9. Selected Topics on Gabor Frames
measure. We leave the slight modifications in the case Д = Q to the reade and assume that Q \ Д 0. By choosing / 6 L2(R) such that Zaf = Xq\& it follows that (f,EmbTnag) = 0 for all m,n e Z, so {EmbTnag}minEZ js incomplete in L2(R).
For the rest of the proof we note that for any F e L2(Q) we have FZag e Ll(Q). Since	is an orthonormal basis for L2(Q),
/•______________2
53	Е(^п,п)Еад)l2(Q)\	— 53 / E(m,nj
тп,п€%	тп,пе% ®
\FZ^g\2.	(9.21J
(ii)-(iv) now follows by a standard argument (Exercise 9.4), yielding e.g., that
[ \FZ^\2 < В ||^llb(Q), *F G L2(Q) \Zag\2 < B, a.e. (9.22) Q
□
Lemma 9.4.2 and Proposition 9.4.3 put restrictions on the functions g for which {ЕтЬТпад}т1пе1 can be a Riesz basis for ab = 1. For example, the Gaussian g(x) = e~x /2 has a continuous Zak transform with a zero, which by Proposition 9.4.3 implies that {^mb^na^}m,nez can not be a Riesz basis.
For the rest of this section we consider a rationally oversampled Gabor system {EmbTnag}minez, i.e., we assume that
ab G Q, ab = - with 1 < p < q.
Q
We always choose p, q such that gcd(p, q) = 1. We state results by Zibulski and Zeevi, resp. Janssen. The references for further information and proofs are [292], [185], [186], and [187].
In the special case considered here, the Zibulski-Zeevi matrix associated with a Gabor system {EmbTnag}minez is a useful tool. It is defined as
$9(t, i/) = p~^ ((Zi g)(t — Z-, и 4- -Й	, a.e.
V b	q p //c=O,...,p-l;Z=O,...,q-l
In terms of this matrix, {Em6Tnag}m>nGz is a Bessel sequence with bound В if and only if	are for a.e. t,i/ G [0,1[ bounded linear mappings
of C7 into Cp, with norms at most В ъ. If we do not need the information about a specific Bessel bound, this result has a nice formulation:
Theorem 9.4.4 Let the setup be as above. A rationally oversampled Gabor system {EmbTnag}m,nez Is a Bessel sequence if and only if there exists a constant C > 0 such that
|Zi <?(t, i/)| < C, a.e. t,i/G[0,1[.
9.5 Tight Gabor frames 219
eNote that Theorem 9.4.4 generalizes Proposition 9.4.3(h) to the case of ational oversampling.
Some of the results proved in [187] and [292] are (still assuming rationally oversampling):
• if {ЕтьТпад} is a Bessel sequence, then the frame operator is represented by
v) = $9(t, v) ^))	a.e. i,v.
•	When А, В > 0 are given, {Em6^na^}m,nGZ is a frame with frame bounds A, В if and only if
Al < $9(t, i/)	i/)^ <BI, a.e. t,u.
Here I denotes the identity operator on O7.
•	Two Bessel systems {EmbTnag}minEz and {EmbTnah}minez are dual if and only if for a.e. а, и e [0,1[ and all к = 0,... , p — 1,
i s-1	_________________
-	(t - lp/q, v + к/p) (Zih) (t - Ip/q, v) = 6kfi.
p 1=0
Let us finally mention that Janssen in [187] proved that if {EmbTnag}m,nez is a frame and satisfies condition (A), then S~lg satisfies condition (A) with the same parameters.
One might wonder if the assumption of {EmbTnap}m>n6z being rationally oversampled is for technical reasons or in the nature of Gabor analysis. It turns out to be the second option. Already in Section 9.1 in the discussion of condition (A) we saw a case where a rational parameter is essential, and in the latest development of the theory there are several results for rationally oversampled systems which do not hold in the general case. Without going into details, the rationality opens for the use of Banach algebra techniques which are not available in the general case; see [158].
9.5 Tight Gabor frames
In applications of frames it is inconvenient that the frame decomposition, stated in Theorem 5.1.6, requires inversion of the frame operator. As we have seen, one way of avoiding the problem is to consider tight frames. Fortunately, tight Gabor frames exist and have been characterized: we saw one characterization in Theorem 8.7.2, and will present others in Theorem 9.5.2.
220	9. Selected Topics on Gabor Frames
Lemma 9.5.1 Let g,h e L2(R) and a,b > 0 be given. Fix nEZ. Then
(i)	h is orthogonal to EmbTnag for all m ф 0 if and only if there is a constant C so that
h(rc — k/b)g(x — k/b — na) = C, a.e. fcez
(ii)	h is orthogonal to EmbTnag for all m E Z if and only if
h(x — k/b)g(x — k/b — na) = 0, a.e. kez
Proof. Lemma 8.4.2 shows that for any m,n E Z, rl/b	_______________
[h, EmbTnag) = V h(x - k/b)g(x - k/b - nty-1’'”*1 dx.
Jo k&
Since {b1/2e2’rlml’}mSz is an orthonormal basis for L2(0, l/Ь), it follows that for a given nEZ,
(h, EmbTnag) =0, Vm E Z
$
h(x — k/b)g(x — k/b — na) = 0, a.e. x E [0,1/6]. kez
Since x h-> h(x - k/b)g(x — k/b - na) is periodic with period 1/b this proves (ii). The statement (i) also follows from the expression for (Л, EmbTnag).	П
We now state equivalent conditions for {EmbTnag}m^nez being a tight frame. The equivalence (i)o(ii) below actually follows from the characterization in Theorem 8.7.2 of shift-invariant systems generating tight frames (Exercise 8.8), but we include a direct proof.
Theorem 9.5.2 Let g E L2(IR) and a,b > 0 be given. The following are equivalent:
(i)	{ЕтьТпад}™^^ is a tight frame for L2(R) with frame bound A = 1.
(ii)	We have:
(a)	G(x) := £n6Z |p(i - na)|2 = b, a.e.,
(b)	Gk(x) :=	g(x — na)g(x — na — k/b) = 0, a.e. for all к 0.
(iii)	g ± Em/aTn/bg for all (m,n)	(0,0), and ||#||2 = ab.
(iv)	{Em/aTn/bg}m,nez is an orthogonal sequence and ||(?||2 = ab.
Moreover, when at least one of (i)-(iv) holds, {Б^Т^д}™,^ is an orthonormal basis for L2(R) if and only if ||#|| = 1.
9.5 Tight Gabor frames 221
proof. (i)=> (ii): Assume {EmbTnag}m,nez is a tight frame for L2(K) with frame bound A = 1. For any function f e L2(IK) which is supported on an interval of length at most 1/b we see that f(x)f(x — k/b) = 0 for all x G К and all к E Z \ {0}. Via Lemma 8.4.3,
E \f(x)\2 dx =	£ К/,ЕтЛа5)|2
m,nEZ
1 /*°°
= т/ ш-ei2 £ ы®-na)i2dx
J-°°	n&
1 /*°°
= bj_	dx-
Since this equality holds for all f € L2(Z), for any interval I of length at most 1/b, it follows that G(x) = b for a.e. x G R Therefore
£ |(/A«|2 = | Г |/«G(z) dx
m,na
for all functions f G L2(R). Using Lemma 8.4.3 again, we have for all bounded, compactly supported f G L2(K),
1	foo_____	_____________
-	/ f(x)f(x - k/b) g(x — na)g(x — na — k/b)dx = 0.
b	nez
A change of variable shows that the contribution in the above sum arising from any value of к is the complex conjugate of the contribution from the value —k. Therefore
o° / roo ________ ____________ ________________________ \
y^Rel / f(x)f(x — k/b) У2 ~ na)g(x — na — k/b)dx I =0. (9.23) k=i V~°°	nez	/
Now fix kQ > 1 and let I be any interval in К of length at most 1 /b. Define a function f G L2(R) by:
f (x) = e-^gG^0(z)? for all x e ц
and f(x — k0/b) = 1 for all x G I and f(x) = 0, otherwise. Then, by (9.23),
°°	/ roo _____ _____________________________________ \
0 = У^ Re j / f(x)f(x — k/b)^^g(x — na)g(x — na — k/b)dx\ k=i	nEZ	/
= Re f /* f(x)f(x - ko/b)Gko(x) dx] = f |Grfco(rr)| dx. \J-oo	/ Ji
It follows that Gko (x) = 0, a.e. on I. Since I was an arbitrary interval of length at most 1/b, we conclude that Gko = 0. A direct computation shows that
G-k0(x) = Gko(x + k0/b) = 0,
222
9. Selected Topics on Gabor Frames
and we have proved statement (b) in (ii) for all к 0.
(ii)=> (i): The assumptions in (ii) imply, again by Lemma 8.4.3, that for all bounded, compactly supported / £ L2(IR),
1 Г°°
E la.^Mi2 = a/ i/«Ei^-na)i2dx
m,n€Z	00	nGZ
= ll/ll2-
Since the bounded compactly supported functions are dense in L2 (R), the conclusion follows by Lemma 5.1.7.
(ii)<=> (iii): By Lemma 9.5.1 (ii), the statement (b) in (ii) is equivalent to g ± Em/aTn/bg for all m e Z,n ф 0. Using Lemma 9.5.l(i) with n = 0, the function G is constant if and only if g ± Em/ag for all m 0; and if this is the case, the relationship between ||g||2 and G(x) follows from
Zoo	pa
|p(x)|2 dx =	/ ^\g(x-na)\2 dx
•°°	nez
= /* G(x) dx
Jo = aG(x).
(iii)o (iv): This follows from the observation that for all к 6 Z,
(Em/aTn/bg, EkiaTg/bg') = е27гг ° b (ff, Ek-™ Tt-^g).
For the final part of the theorem, we just observe that if {EmbTnag}mine^ is a tight frame with frame bound 1, then for any (m',n') 6 Z2,
I\EmibTniag\|	= \{Em/bTn/ag1 EmbTnag)|
m,nEl
—	|4 4-	\(EmibTniag^ EmbTnag)\ .
,n')
If ||p|| = 1 it follows from here that {EmbTnag}minEz is an orthonormal system.	□
9.6 The lattice parameters
Whether a Gabor system {EmbTnag}minEz forms a frame or not depends on a complicated interplay between the parameters a, b and the function g. Even by fixing the function g E L2(IR), the frame condition is in general very sensitive towards the choice of a,b. Recall e.g., Janssens results for the characteristic function in Section 8.6: taking g = X[o,7/4]> they show for example that
9.6 The lattice parameters 223
•	{EmTnag}mtnEz is a frame if a < 1 and a £ Q
•	{ЕтТпад}т^пЕх is not a frame if a e {|j}.
The possibility of such a strange behaviour for functions in L2(K) is one of the reasons for considering functions g belonging to the Wiener space W: in Proposition 8.5.2 we proved that in this case {ETObTnap}m,nez is at least a Bessel sequence for any choice of a, b > 0. Furthermore, a very reasonable condition in Proposition 8.5.3 shows that by fixing a > 0, we obtain a frame for all sufficiently small values of b > 0.
For a function g e L2(IR) generating a Bessel sequence {EmbTnag}minEz for some a, b > 0, a result by Feichtinger and Janssen [133] shows that the Bessel property is at least preserved if a, b are replaced by rationally related parameters:
Lemma 9.6.1 Let g 6 L2(1R) and a,b > 0 be given. If {EmbTnag}m,nez is a Bessel sequence, then {Embr/sTnap/qg}mynez is a Bessel sequence for any r,s,p,q £ N. Denoting the optimal Bessel bound associated with the parameters a,b by B(a,b),
B{ap!q,brIs) <qsB(a,b) < pqrsB(ap/q,br/s).
(9.24)
Proof. Let s > 0 be any integer. Note that if j runs through 0,..., s —1 and m runs through Z, then ms 4- j runs through Z. Therefore, for f E L2(K),
E \(f 5 -^mbr / s^nap/q9^) | m,nEZ
EE E	Eb(ms+j)r / s^a (nq+l)p/q9)\2
j—0 1=0 m,nEZ
EE E \(E—bjr/sT—alp/q f •) EmbrTnapg)\ j=0 1=0 m,n£Z
EE E I {E—bjr/s^—alp/qfi EmbTnag) I j = 0 1=0 m,n£Z
|2
j—0 1=0
qSB(a,b)\\f\\2.
Thus {Embr/s'Enap/q9}7n,7ie'i is a Bessel sequence with bound
B(ap/q,br/s) < qsB(a,b).
224
9. Selected Topics on Gabor Frames
Using this result with a, b replaced by ap/q^br/s and p/q^r/s replaced by q/p, s/r leads to
B(a,b) = В
\ qp s rJ
< prB(ap/q, br/s).
□
For (9.24) to hold it is crucial that we are speaking about the optimal bounds. We also note that the same Gabor system {Embr/sTnap/qg}mtnez can appear by different choices of r, s,p, q; in fact, having one choice we obtain another if we multiply all four numbers with the same к 6 N. For the estimate (9.24) to be interesting it is important to choose the smallest possible parameters, i.e., we take r,s,p,q such that
gcd(r,s) = gcd(p,q) = 1.
As a special case of Lemma 9.6.1 we note that if {EmTng}minez is a Bessel sequence, then {Emr/sTnp/qg}m,nEz is a Bessel sequence for all r, s,p, Q e N. The points {(p/<7,r/s)}r^PiqEN are dense in ]0,oo[x]0,oo[, so it is natural to ask if {Emb^na^}m,nez automatically is a Bessel sequence for all a, b > 0 in this case. Feichtinger and Janssen proved that the answer is no:
Proposition 9.6.2 Let a >0 be any irrational number. Then there exists a function g 6 С°°(К) П L2(]R) with | supp(g) | < 1, for which
(i) {EmbTnag}m is a Bessel sequence for all rational a,b > 0.
(ii) {EmbTnacg}m,nEZ is not a Bessel sequence if c > 0 is rational, regardless of the choice of b > 0.
Proof. By Lemma 9.6.1 we obtain (i) if we construct g such that is a Bessel sequence; by Proposition 9.4.3 this is the case if the Zak transform Zg is bounded, and a sufficient condition for this is that
sup У2 \g(x 4- n)| < oo.	(9.25)
^[o,i] nGZ
Also, the negative conclusion in (ii) is by Proposition 8.3.2 and Lemma 9.6.1 obtained if
sup |р(т 4-no)|2 = oo.	(9.26)
^[o,i] nGZ
In fact, in this case {EmbTnag}minez is not a Bessel sequence, and therefore is not a Bessel sequence either when c G Q. We will now construct a smooth function, which satisfies (9.25) and (9.26).
We start with the given irrational number a, and observe that the set {na — LnaJ}^=i is dense in ]0,1[ (Exercise 9.5). We now construct a sequence of mutually disjoint intervals	contained in ]0,1[ <as follows.
9.6 The lattice parameters 225
The interval Zi is defined by
Zi =]a- |aj -61,a - |aj 4-сЦ,
where 6i < a/2 is chosen so small that Ii is in fact contained in ]0,l[. Since the set {na - |naj }^=1 is dense in ]0,1[, we can now find an interval /2 c]0,1[ \ A of the form
Z2 =]n2o - [n2aj - 62,n2a - |n2aj 4- e2[.
In fact, we can take n2 > 1 such that n2a — [n2aj e]0,1[ \ Zi and then choose c2 < a/2 so small that
]n2o - |n2aj - c2,n2ct - [n2aj 4- e2[c]0,1[ \ Zi.
Continuing this process inductively we obtain the desired intervals {Zfc}^_1, where each interval Ik has the form
Ik =]nfca - |nfcaj - ек,пка - |nfcaj 4- 6fc[;
we choose the sequence	to be increasing, and we take ek < a/2.
For each к G N we now let Jk denote the middle third of Ik and choose a smooth “plateau” function gk supported on Ik and satisfying
0 < 9k < 1 on Ik, gk = 1 on Jk.
Finally, let
oo
P(z) :=	- |n*aj).
/c=l
The disjoint support of the functions gk implies that g is well defined, smooth, and has a support with measure at most 1. This also yields (9.25), so we only have to verify (9.26). By definition,
E |р(ж + na)|2 = УЗ ^gktx + na- Ln*aJ) n£Z	n£Z k-1
By throwing positive terms away in the sum on the right-hand side, we see that for any К G N,
К oo
52 1р(ж+nQ)i2 > EE gk(x+nja- [гна]) nez	j=i k=i
For all x G П£=1] “ 6fc/3,6fc/3[ and j =
oo
^9k(x + nja - LnfcaJ)
k=l
> gj(x 4- nja — [njaj) = 1;
therefore
+ na)|2 > K.
226
9. Selected Topics on Gabor Frames
Since К G N is arbitrary, this implies (9.26).	□
Another amazing example in [133] is a function g for which
(i) {EmbTnag}minEz is a frame for all a = к e N and b e]0,1[, and
(ii) {EmbTnag}m>nEz is never a frame when a =	k,£ e N and b G]0, Г.
The results by Feichtinger and Janssen show that one has to be very careful when dealing with Gabor systems for general functions. For example, numerical calculations will always lead to round-off errors, which can change the properties of a Gabor system drastically. The way to avoid the problem is to use a class of “well-behaving” generators g E L2(JR) for which the undesired phenomena do not appear. It has been known for some years that when g G L2(]R) \ {0} is in the Feichtinger algebra So, then there exist constants aQ,bo > 0 such that {EmbTnag}minEz is a frame for L2(]R) for all a G]0, a0], b G]0, bo]', see Corollary 17.2.6. More recently, Feichtinger and Kaiblinger proved in [134] that the triples (g, a, b) in 5o x R+ x K+ which generate Gabor frames is open with respect to the product topology; in particular, for a fixed g E So the set of points (a, 6) G]0, oo[x]0, oo[ for which {EmbTnag}m,nei is a frame is open. Thus, the use of generators in 50 will lead to Gabor systems which are less sensitive towards round-off errors.
9.7 Irregular Gabor systems
Till now we have only considered Gabor systems of the special form {ЕтЬТпад}т,пЕ%, i.e., time-frequency shifts of the function g along a lattice {(na, mb)}m>nGz- By replacing the lattice with a countable sequence of points {(/zn, Лп)}п€/ С K2 we obtain a more general Gabor system of the form
= {e2niXnXg(x - цп)}пе1.	(9.27)
We call (9.27) an irregular Gabor system.
The analysis of irregular Gabor systems is complicated, and the theory is not very well developed. Especially the general case described here, where no structure on the sequence {(/xn, An)}nG/ is assumed, causes difficulties. There are, however, some types of irregular Gabor systems which somehow lie between the systems {E\nT^ng}nEi and the regular systems {EmbTnag}minEz. In fact, if we consider two countable sequences {Mn}nez, {Am}m€z C R, then the Gabor system
{EXmT.ng}m,nez	(9.28)
still has some kind of lattice structure: if {/in}nez5 {Am}mez are increasing and /zn,Am -> ±oo for m,n -» ±oo, this Gabor system also splits the
9.7 Irregular Gabor systems
227
time-frequency plane IR2 into boxes, but with varying size. We encourage the reader to make a picture, based on for example цп = Xn — n2lnL If we further assume that Xm = mb for some b > 0 (or fjLn = an) we are “close” to the regular case. This even holds in a more precise sense: several results for Gabor systems {ЕтъТпад} m,n£Z can be extended to the case where only one of the sequences {mb}mGz, {na}nez is replaced by an irregular sequence. Theorem 8.4.4 is an example of a result which can be generalized this way; we refer to [185] and [78] for different approaches.
An important result was proved by Sun and Zhou in 2001 [274]. It is based on the assumption that g as well as the function x i-> xg(x) belongs to the Sobolev space #i(1R). In order to avoid cumbersome notation we allow us to write the latter assumption as xg(x) 6 Я1(К):
Theorem 9.7.1 Assume that g, xg(x) € /fi(lR) \ {0} and that a,b > 0 are chosen such that
Д := “llsHI + 46	+ — Ikff'(z) + p(x)|| < ||p||.	(9.29)
7Г	7Г
Let {(/^ттт,,/!, ^771,71Tn,zibe chosen such that
Xm,n) 6 [na, (n 4- l)a[x[mb, (m 4-1)5[, Vm, n G Z. (9.30)
Then {Exm nT^m>ng}m,nez a frame for L2(JR) with frame bounds
-L(||p|| - A)2, l(||p|| + A)2. ab	ab
To avoid confusion we note that all norms in (9.29) are L2(IR)-norms. We will not prove Theorem 9.7.1 (it can be considered as an explicit version of Corollary 17.2.6), but it is worth discussing the assumption (9.30). When a, b > 0 are given, the boxes
[na, (n 4- l)a[x[mb, (m 4- 1)6[, m,nEZ,
form a partition of JR2 into disjoint sets. In words, Theorem 9.7.1 says that by taking a, b small enough and picking exactly one point from each box, the associated time-frequency shifts of g will form a frame for L2(IR). Thus Theorem 9.7.1 can be considered as a density result, saying that if the points {(Мтп,7i, Am)n)}m>nGz are “dense enough in IR2 but not too close”, then they generate a frame.
On the other hand one can also consider Theorem 9.7.1 as a perturbation result. In fact, the conditions imply that {EmbTnag}m,nez itself is a frame; now the natural interpretation of condition (9.30) is that {£A„,.3k.nj}m,nez is a frame if the points {(Mm.n, Am,n)}m,nez are sufficiently close to {(na,mb)}m nSz- We return to perturbation of Gabor frames in Section 15.4.
Irregular Gabor systems were already considered around 1990 in a series of papers by Feichtinger and Grochenig [128], [154]; this was in a more
228
9. Selected Topics on Gabor Frames
general context, to which we return in Section 17.2. The first direct approach to irregular Gabor systems in L2(R) was by Grochenig [155] in 1993. Around the same time Ramanathan and Steger [239] studied completeness properties of irregular Gabor systems in terms of the Beurling densities of {(дп, An)}, defined in (7.1). In particular they proved that the density must be exactly 1 in order for {ExnT^ng} to be a Riesz basis. Later Christensen, Deng and Heil [77] proved that for {Е\пТ^пд} to be a frame, it is necessary that {(/zn, An)} be relatively separated and that D~({(/zn, An)}) > 1. For a regular Gabor system {EmbTnag}m,nEz the latter assumption corresponds exactly to the condition ab < 1. As noted before,{ЕтъТпад}т>пе% is not complete in L2(R) if ab > 1. One could therefore expect that {E\nT^ng} must be incomplete whenever	An)}) < 1, and this appears as a
conjecture in [239]. However, Benedetto, Heil and Walnut [20] have shown that this is false. We will discuss this in some detail, but leave most of the calculations to the reader (Exercise 9.6). The construction is based on a result by Landau [204]:
Lemma 9.7.2 Let 5 G]0, |[ and К G N be given and define
K—l q	-
J = U]n-(2-<5)’n+2~'5[-	(9-31)
n=0
Then, for any e > 0 there exists a symmetric real sequence {Am}mGz for which |Am — m| < c for all m EZ and such that {е2'!ггХтХ}те1 is complete in C(I) with respect to the || • Цоо-norm.
Recall from Section 7.1 that a sequence {Afc}kGz for which the upper and lower Beurling densities coincide is said to have a Beurling density, which equals the upper and lower densities and is denoted by D({Afc}fcez)«
The exact statement of the result by Benedetto et al. is as follows:
Proposition 9.7.3 Given an arbitrary e > 0, there exists a sequence {(Мп, An)} C IR.2 and a function g 6 L2(IR) such that
(i) D({(/in,An)})<6;
(ii) {Е\пТ^пд} is complete in L2(IR).
Proof. In order to apply Lemma 9.7.2, we fix c G]0, l/2[ and 6 G]0, l/4[. For a given К 6 N we use the notation in Lemma 9.7.2 and choose a real sequence {Am}mGz such that |Am — m| < e and {е27ггЛт'1Х }mGz is complete in C(Z). Let
F • {(ZCn, Am)}m)nGz U {(A"?!	2?
The Beurling densities of {Am}mGZ and {Kn}na in R are
£({AraUz) = l, Л({Кгг}„е2) = -L;	(9.32)
9.7 Irregular Gabor systems 229
it follows from here that
2
Р(Г) =	(9.33)
We now prove that the irregular Gabor system associated to A and the function g = Xi, i.e.,
{-^А1,Лк'пА.'/}т.пег U {E\m Tj^n+ 1 Xi }m,n£Z?	(9.34)
is complete in L2(IR). Suppose that f 6 L2(IR) is orthogonal to all the functions in (9.34). Considering an arbitrary fixed value of n G Z, we have
Zoo
f(x)e~2*iXmXxi(x — Kn)dx
-oo
__ 2ттгАгп К n
le 2niXmXdx.
Since {е~21ггХтХ}тЕ% is complete in L2(Z), it follows that f(x 4- Kn) = 0 for a.e. x G I. This holds for all n G Z. A similar argument gives that f(x 4- Kn 4- 1/2) = 0 for a.e. x 6 I and all n G Z. By the choice of I in (9.31),
R = |U(7 + 7<n)) j|U(/ + /fn+l) \n£Z	/ \n£Z
(9.35)
so we conclude that f = 0 a.e. Thus the functions in (9.34) are complete. Since this construction is possible for all К G N, we are done.	□
Let us end this section with a few words about a conjecture by Heil, Ramanathan and Topiwala [170], which we stated already on page 32: it says that if {(/xn,An)} С K2 is a finite set of distinct points and g is a non-zero function in L2(R), then {E\nT^ng} is linearly independent. The conjecture is proved under some extra assumptions in [170]. Later, Linnell [214] was able to prove it in the case where {(/in,An)} is a lattice (or a subset hereof), i.e., for
{(МпЛ)} = {(na,mb)}^m=1.
Thus, finite subsets of a regular Gabor frame {EmbTnag} for L2(R) are linearly independent. The general conjecture is still open.
The linear independence of the elements in a frame {ЕтьТпад}т)Пег has the surprising consequence that the lower frame founds for finite subfamilies {EmbTnag}\m\>\n\<N are forced to go to zero for N -> oo if {EmbTnag}minez is overcomplete:
Theorem 9.7.4 Suppose that ab < 1 and that {Е^ьТпад}™^^ is a frame for L2(IR). Let Ац denote the optimal lower frame bound for
230
9. Selected Topics on Gabor Frames
{ЕтЬ^падУ |m|, |n| < N • Then
An —> 0 as N —> oo.
Proof. By Theorem 8.3.1, the assumption ab < 1 implies that {EmbTnag}m,n(=z is not a Riesz basis. By Linnell’s result, {EmbTnag}minel is linearly independent, so for each N G N, {ЕтЬТпад}\тцп\<м is a (Riesz) basis for its span. The optimal lower frame bound An for {EmbTnag}\m^\n\<N coincides with the optimal lower Riesz bound by Theorem 5.4.1, and the sequence	is decreasing. Now the conclusion
follows by Proposition 6.1.2.	□
9.8 Applications of Gabor frames
There is a large diversity of research fields where Gabor systems and frames play a role. We will mention a few of them as inspiration for further reading and provide a slightly more detailed discussion of an application to pseudodifferential operators.
We have already noticed that Gabor systems appeared already around 1930 in the context of quantum mechanics and that they also find use in the study of molecules [114]. A large diversity of applications appear in the two books [136] and [137], which contain articles by researchers in different fields. Among the applied papers in [136] are
•	Gabor representation and signal detection ( Zeira and Friedlander);
•	Multi-window Gabor schemes in signal and image representation (Zeevi, Zibulski, Porat);
•	Gabor kernels for affine-invariant object recognition (Ben-Arie and Wang);
•	Gabor’s signal expansion in optics (Bastiaans).
From [137] we mention
•	Optimal stochastic approximations and encoding schemes using Weyl-Heisenberg sets (Balan and Daubechies);
•	Orthogonal frequency division multiplexing based on offset-QAM (Bolcskei).
We also note that an application of Gabor frames to noise reduction appeared already in [224].
We now go a little more into detail with an application of Gabor frames to estimation of singular values of compact operators defined via the Weyl correspondence. In general, the singular values of a compact operator L on a Hilbert space H are defined as the eigenvalues of the compact self-adjoint
9.8 Applications of Gabor frames 231
operator (L*L)1/2. Alternatively, when the singular values are arranged decreasingly, the n-th singular value is
sn(L) = inf{||L - F|| : F has finite rank and < n}.	(9.36)
One says that L belongs to the Schatten-von Neumann class Sp, 0 < p < oo, if oo
^Sn(L)”<oo.
n=l
To introduce the operators we will consider, we need the Wigner distribution of f,g G L2(IR), which is defined by
= £° e~2nip^f(x + fyg(x - ^)dp.
Let 5(R) denote the Schwarz space of rapidly decreasing functions on R One can show that if /, g G 5(R), then W^f.g} G 5(IR2). The Weyl correspondence associates to each tempered distribution a G 5'(IR2) a pseudodifferential operator La : 5(R) -» 5'(JR), defined via
(Laf,g) =
In case a corresponds to a function, ZOO POO
/ a^,x)W(f,g)(i,x)d(dx, /)5 £<$(»)•
-oo J — oo
One also writes
Laf(z)= Г Г
It is known that La defines a compact operator on L2(JR) if a G L1(JR). There is a rich literature concerned with estimates of the corresponding singular values. The first appearance of Gabor frames in this context was in the paper [244] by Rochberg and Tachizawa, where they were used to find conditions on a implying that La belongs to a given Schatten-von Neumann class (see also [275], where Wilson bases are used instead of Gabor frames). The same theme was taken up by Heil, Ramanathan and Topiwala in [171] (and more recently by Heil in [169]), and we sketch their main idea here.
The characterization of the singular values in (9.36) indicates how frames can be used to estimate sn(L), even in the general case of an operator L on a general Hilbert space H. In fact, letting {fk}kLi be a frame for H with frame operator S, the finite partial sums of the frame decomposition (5.7) define operators of finite rank. More precisely, for any n E N the bounded operator
n
Fnf = YSf’S^Mfk
k=l
232	9. Selected Topics on Gabor Frames
has rank at most n, so
sn(L) < ||L - Lf’n-ill-
The actual technique in [171] is a variation of this idea. In fact, the authors approximate La by operators of the form Lan, where an = Fna and {A}£i is a Gabor frame for L2(K). With this alternative approach it is not immediately clear that Lan has finite rank (and how large it is) but the authors provide the necessary estimates.
From the above short description it is not clear why we need to use overcomplete frames in this application. Why not just use an orthonormal basis? The answer is that for technical reasons one needs the generator g of the Gabor frame as well as its Fourier transform to be well localized, i.e., to decay fast. The exact condition collides with the Balian-Low theorem, and can not be combined with g generating a basis. For this reason we have to use an overcomplete frame, where well-localized generators are possible; in fact, the authors use a Gaussian. The over completeness of the used frame by itself is not needed.
9.9 Wilson bases
Already in Chapter 4 we have discussed some of the limitations on the function g if we want {EmbTnag}mfn^z to be a Riesz basis for L2(R). In particular, the Balian-Low theorem shows that g can not be well localized in both time and frequency. However, these properties can very well be combined with {EmbTnag}m,nez being a frame, (the Gaussian is a concrete example) and this is just one motivation for the study of Gabor frames.
Daubechies, Jaffard and Journe [109] proposed in 1991 another way to circumvent the Balian-Low theorem. They proved that if one is ready to give up the Gabor structure, it is possible to obtain a well-localized orthonormal basis: more precisely, if g G L2(JR) is chosen such that g is real-valued and {EmTn/zg}m,ne% is a frame with bounds A = В = 2, then the collection of functions {'ipi,k}i>o,ke% defined by
g(x — k)	for I = 0,
= V%g(x — k/2) cos(2ttZz) for I > 0, к + I even,
^\/2g(x — k/2) sin(27rZz) for I > 0, к + Z odd
constitute an orthonormal basis for L2(R). A basis of the form
is called a Wilson basis. In terms of the modulation operators and translation operators,
Е$Ткд	for I = 0,
'Фцк = <	+ E-tTk/‘2g') for I > 0,k + I even,
^^'(EiTk/z.Q — E-iTk/i9) for I > 0, к 4- I odd,
9.10 Exercises 233
i.e., the functions in the Wilson basis consist of linear combinations of the functions in the Gabor system {EmTn/2g}m,neZ’ By choosing g such that the above mentioned conditions are satisfied and
( [ |zp(z)|2<fo ) ( [ l7P(7)|2rf7 ) < oo, W—oo	/ W — oo	/
we have obtained an orthonormal basis circumventing the Balian-Low theorem. We refer to [9], [20] and [109] for more information, especially to [109] for a construction of a suitable function g.
Observe that the important feature of the system {V>z,fch>o,fcGZ is that it is an orthonormal basis. It is not complicated to construct frames with a similar structure:
Proposition 9.9.1 Let g e L2(R) and a,b> 0 be given, and suppose that {EmbTnag}mine% is a frame with upper bound B. Then the functions
{g(x — na)}nez U {cos(27rmbx)g(x — na), sin(27rmbx)g(x — na)}mE^nEz
constitute a frame for L2(R) with upper bound B.
For the proof, one can check (Exercise 9.9) that the pre-frame operator corresponding to the functions in Proposition 9.9.1 is bounded and surjective, so the result follows by Theorem 5.5.1.
Riesz bases of Wilson type are investigated by Trebels and Steidl in [278]. Overcomplete Wilson expansions are also studied in [36]. For surveys on Wilson bases we refer to [29], [28].
9.10 Exercises
9.1	This exercise concerns condition (R) and its relationship to Lebesgue points.
(i)	Assume that g 6 L2(IR) satisfies condition (R). Show that all integers are Lebesgue points for g.
(ii)	Assume that g is a bounded compactly supported function for which every integer is a Lebesgue point. Show that g satisfies condition (R).
(iii)	Prove via (ii) that condition (R) is satisfied on a dense subset of L2(R).
(iv)	Prove that the Gaussian g(x) = e~ix2 satisfies condition (R).
9.2	Find a Gabor system {ЕтьТпад}т,пе% which satisfies condition (UCC) without the Bessel condition, and which does not satisfy condition (CC).
234
9. Selected Topics on Gabor Frames
9.3	Prove (i) and (ii) in Lemma 9.4.2.
9.4	Complete the proof of Proposition 9.4.3 by proving (9.22) and the similar statement for the lower bound.
9.5	Let a £ Q and prove that the set {na — LnaJ}^=i dense in [0,1] (use e.g., the following result attributed to Weyl: if f is a continuous 1-periodic function, then /J f(x)dx = limyv-^oo у 52n=i f(na) \
9.6	In this exercise we ask the reader to provide some of the details in the proof of Proposition 9.7.3.
(i)	Verify (9.32) and (9.33).
(ii)	Choose the sequence {Am}mez as in Lemma 9.7.2. Prove that {е27ггЛт }mGZ is complete in L2(Z).
(iii)	Verify (9.35).
9.7	Let Q = [0, l[x[0,1[. Prove that L2(Q) C LX(Q), and find a function f e LJ(Q) which does not belong to L2(Q).
9.8	Describe how a sequence {em>n}m)nGz can be reindexed as
9.9	Prove Proposition 9.9.1.
ю
Gabor Frames in £b'Z)
Every numerical calculation with functions in L2(K) will involve a discrete model, where all calculations are done with (finite) sequences in ^2(Z). Therefore it is important to know that certain conditions on a Gabor frame {EmbTnag}m>nez in fact imply that we can construct a frame for ^2(Z) having a similar structure. The relevant conditions were discovered by Janssen, and the main part of this chapter will deal with his results.
One can also consider frames in £2(Z) with a Gabor-like structure without referring to frames in L2(K). The theory for these frames is very similar to the Gabor theory in L2(K) and will not be discussed in detail.
10.1	Translation and modulation on £2(Z)
For a sequence g G ^2(Z) we denote the j-th coordinate by g(j). Thus
# = (... ,p(—l),p(0),p(l),...).
We now want to define the modulation operator Eb,b GK, on ^2(Z). That is, given g 6 ^2(Z) we want Ebg to be a sequence in ^2(Z); we define it to be the sequence whose j-th coordinate is
Ebg(j):=e2^g(j).	(10.1)
Even though the definition of Eb makes sense for all b G К we will only use modulations of the form Em/Mi where M G N is fixed and m G Z. In the terminology used for Gabor systems in L2(K) this corresponds to having the modulation parameter equal to 1/M. There is, however, one
236	10. Gabor Frames in £2(Z)
important difference between the two settings: in the L2(K)-setting, modulation operators with different parameters are necessarily different, but this is not the case in the discrete setting discussed here. In fact, with the definition (10.1),
Ein = Ein+k for all к e Z.
Therefore {Ет/Мд}тЕ% can not be a Bessel sequence in ^2(Z) unless g = 0. For this reason we will only consider modulations £т/м with m,..., M — l, We now introduce the translation operator on ^2(Z). Given n G Z and g 6 ^2(Z) we let Tng be the sequence in £2(Z) whose j-th coordinate is
ГпрО) = g(j - n).	(Ю.2)
The discrete Gabor system generated by a sequence g 6 <?2(Z) and with modulation parameter 1/M and translation parameter ЛГ, (M,N 6 N) is now defined as the family of sequences {Ет/мТп^д}пЕг,т=о,...,м-1\ specifically, Ет/мТпмд is the sequence in ^2(Z) whose j-th coordinate is
Em/MTnNg(j) = e2^Mg(j - nN).
Many results for Gabor systems in L2(JR) have analog counterparts for Gabor systems in ^2(Z), with similar proofs. For example, a necessary condition for {Ет^мТп^д}пЕ2,т=о,...,м-1 to be a frame for £2(Z) is that < 1; and if {Em/MTnNg}nEztm=o,...,M-i is a frame, it is a Riesz basis if and only if M = N. We will not go into the general theory (see e.g., [102], [103], [282]) but concentrate on a method for constructing a Gabor frame for ^2(Z) based on a Gabor frame for L2(R).
10.2	Discrete Gabor systems through sampling
Janssen proved in [189] that there is a natural way to obtain discrete Gabor frames via Gabor frames for L2(K) through sampling. We present some of his results here.
The starting point is a Gabor system for L2(R); we assume it to have the form {Ет/МТпкд}т>пЕ%, where g 6 L2(IR) and M,N G N. The first question is how one can construct sequences in £2(Z) based on the Gabor system in L2(]R). If we assume that g is continuous, then we can easily obtain sequences indexed by Z by sampling. That is, for each m,n E Z we consider the sequence
{Em/MTnNg(J)}jEZ = {e2^m'Mg(j -nN)}ja.	(10.3)
The basic idea by Janssen is to ask for conditions such that the family of all the sequences in (10.3), where m = 0,	— l,n G Z, constitute a frame
for £2(Z). The first point is to ensure that the sequences in (10.3) do in fact belong to £2(Z). The condition given in Lemma 10.2.1 will guarantee this.
10.2 Discrete Gabor systems through sampling 237
For Gabor systems {Ет//МТпмд}т,пЕ1 in L2(]R) where g G L2(JR) is not assumed to be continuous, we have to be careful with the meaning of sampling. By definition, L2(R) consists of equivalence classes of functions which are identical almost everywhere, so point evaluations do not immediately make sense as for continuous functions. So when we speak about sampling a function g G L2(R) we really mean that we evaluate a specific representative for the considered equivalence class. We will generally replace the continuity condition by the weaker requirement that the chosen representative has all integers among its Lebesgue points.
Given a function f G L2(K) we now define the discrete sequence
fD = {/(j)W.
With this notation, we can consider the sequence Em^MTnN(fD), obtained by letting the discrete Gabor system act on the sequence fD; or we can consider the discrete sequence (Em/MTnNf)D, obtained by sampling of the function EmfMTnNf G L2(R); the two procedures lead to the same outcome, and we simply write
Em/MTnNfD =	- n^}je2.	(10.4)
The first result gives a condition on the Gabor system {Ет/мТпкд}т,па in L2(K) which implies that the discrete time-frequency shifts of gD belong to £2(Z).
Lemma 10.2.1 Let g G L2(K) and M,N G N be given. Assume that g contains all the integers among its Lebesgue points and that {Em/MTnNg}m,nez is a Bessel sequence in L2(JR). Then
Eisb)i < jez
In particular, Em/MTnNgD G £2(Z) for all m,n G Z.
Proof. Let j G Z and б > 0 be given. Letting В denote an upper frame bound for {Em/MTnNg}mtnEz, we know from Proposition 8.3.2 that
\g(x 4- nAQ|2 < a.e. x G R	(10.5)
nez
In particular, g is essentially bounded. Now let j G Z. The assumption that j is a Lebesgue point combined with (10.5) shows that |p(j)| < Assuming that g is real-valued, it follows that
I ls0')|2 - ls(j + z)|2 I = |(p(j) + p(j+a:))(p(j)-p(j + a:))|
< IsO) “ 9(3 + z)|, a.e. x G R.
238	10. Gabor Frames in £2(Z)
Using that j is assumed to be a Lebesgue point for g,
Ш12--/	!<?(* +j)l2<&
6 J—e/2
1 f6/2
-	(Is(»|2-|p(®+j)|2)<ir
6 J-e/2
I TD i /*с/2
-> 0 as e -> 0.
We conclude that
1 f6/2
|p(j)|2 = lim- /	\g(x+j)\2dx,\/jtZ.
'-*> t J-,/1
(10.6)
We have derived this under the assumption that g is real-valued, but it now follows that (10.6) also holds for complex-valued functions. In the rest of the proof we can proceed without assuming g to be real-valued. Using Fatou’s lemma on X = Z and the counting measure, followed by an application of (Ю.5),
£2ip(j)i2 =	[	ip^+j)!2^
____ 1 fe/2
< limmf^-/	|p(a; + ;)|2da;
v-—1	f6/2
= lim inf ZL_ / 12	}'+пЛГ)12<£г
j=0 e ^-e/2 nGZ
BN
- ~M~'
This proves that gD G ^2(Z); now the lemma follows because the operators
Em/M and TnN on £2(Z) are norm-preserving.	□
The next lemma is an important step from Gabor systems in L2(JR) to Gabor systems in £2(Z). It contains an identity involving functions in £2(JR), which “approaches discrete sequences” for small values of c:
Lemma 10.2.2 Let g e L2(JR) and M,N О be given, and assume that {Ет/мТпмдУтП'Пея is a Bessel sequence in L2(K). Given e e]0, |[, let 
„ 1
6 = ;Х]-4<.И-
Consider a finite linear combination of translates of 5е,
r=^CiT^-	(10.7)
j
10.2 Discrete Gabor systems through sampling
239
Then
E \{f(,Em/MTnNg}\2 тп,п€%.
“h 3 'j^m/M^nN9(.^ “Ь k^dx.
proof. First, we use the definition of fe to write
E \{f(,Em/MTnNg}\2
m,nEZ
= У ^CjCk{Tj6\Em/MTnNg}{Em/MTnNg,Tk6€)
ТП,ПЕ.1 j,k
M-l
— 5 У У У cjck(Tj8 ,Ei+rn/MTnNg)(Ei+m/MTnNgiTk6 ). nEZ 771=0 /£Z j,k
Now, via Lemma 8.4.2,
{Tj8 , Ei+m/MTn^g)
— (E-m/MTj8e, ЕТГпмд)
= Г ( 52 T^x - r)Em/MTnNg{x - r) ) e~2irilxdx, Vez	/
which is the Z-th Fourier coefficient of the 1-periodic function
«j'W = ETiSC(x ~ r}Em/MTnNg(x ~ r) r&
in L2(0,1). Note that for x e [—1/2,1/2],
= 8e(x)Em/MTnNg(x + j)
—	~X] —1 e,|e[(^)^77i/M^P7iN^(:c 4" j)-
Via Lemma 3.7.1,
E^ 7 El+m/M^nN g) [Ei^.m^ M^nN fh'Fkfi ) /ez
—	{^ji^k)
=	/ aj(x)ak{x)dx
- ~2 / Em/MTnNg(x +j)Em/MTnNg(x + k)dxt
and the result follows.	□
240	10. Gabor Frames in £2(Z)
If we impose stronger conditions on g we can obtain a Gabor frame for £2(Z) by sampling of a Gabor frame for L2(K):
Theorem 10.2.3 Let M,N 6 N. Assume that g € L2(K) satisfies condition (R) and that {Ет/мТпмд}т,пе2 is a frame for L2(R) with frame bounds A, В. Then the discrete Gabor system {Em/MTnNgD}nez,m=o,...,M~i is a frame for £2(Z) with frame bounds A, B.
Proof. In order to prove that {Em/MTnN gD}nE%,m=o,...,M-i is a frame for ^2(Z), we consider a finite sequence {ck}kez- Then, for any e e]0, |[ the square of the L2(R)-norm of the function in (10.7) is
urn2 = Jen2-
Applying the frame condition for {Em/MTnN g}m,n£i on fe gives that for а11е€]0,|[,
a£|c>|2 < e £ \{f\EmlMTnN9}\2 < b£|C>|2. j'GZ	m,n€Z	j'GZ
For the proof of Theorem 10.2.3 it is therefore enough to show that
2
M-l	____________
Unnnfe E К/е--Ет/мГ„мр)|2 = E E ’EciEm/MTnN9^ • (Ю-8) m,nEZ	n£Z m=0 j
In fact, if (10.8) is satisfied for all finite sequences {c/Jfcez, then {Em/jvfTn^^D}nGz,m=o,...,M-i satisfies the frame condition in £2(Z) on a dense set, and therefore on ^2(Z) by Lemma 5.1.7.
First we note that
nGZ 771 = 0
У cjEm/M^nNg(j) j
M-l
= EEE Cj Ck Em/M TnNg {j) EmfM Тпяд(к),
T1GZ 771=0 j,k
while by Lemma 10.2.2
e £ \{f\Em/MTnNg}\2 m,nEZ
^-1 ,
TlGZ 772 — 1
10.2 Discrete Gabor systems through sampling 241
Comparing the two expressions we see that (10.8) follows if we can prove that
— f~ ^m/M^nN9^ 4* j)Em/MTnNg(x + k)dx n(=Z 6 J-2е
Em/MTnNg(j)Em/MTnNg{k) as б > 0 n£Z
for all m = 0,..., M — 1 and E Z (recall that the sums over jtk are finite). Now,
I Ет/мТпмg(x 4" j)Em/MTnNg(x 4- k)dx
 J-y
—	-^'m/M^nA,’^(j)-^’m/M-^nN^(^') j
/ IM^nN g(x J^E^/MTnNg(x 4" Aj)
—	Em/MTnNg(j)Em/MTnNg(k)| dx
/ lp(# 4- j — nN)g(x 4- к — nN) — g(j — nN)g(k — nN) I dx
j* i \g(x 4- j — nN) — g(j — n7V)| \g(x 4- к — n7V)| dx
1 fi6 ----------
- IpC? ~ nN)| \g(x + к — nN) — g(k — nN)\dx.
e J-У
It follows that
/ I Ет/МТпмд(х 4" j)Em/MTnNg(x 4~ k)dx
—	Ет/мТп^g(j)Em^TnNg(k)
< - ^2 [ \g(x + j - nN) - g(j - nN)\ |p(x + к -nN)\dx (10.9) 6 nez-'-i'
+-^2 [ lsO ~ n-^)l \g(x + к — nN) — g(k — nN)\dx. (10.10) 6 nez-'-i£
Both (10.9) and (10.10) converge to zero as б -> 0; we give the argument for (10.9). Applying Cauchy-Schwarz’ inequality twice, first on the integral
242
10. Gabor Frames in £2(Z)
and then on the sum,
1 „ _________________ __________
"X / Ых + 3 -nN) -g(J-nN)\ \g(x 4- к -nN)\dx
/ 1€	X 1/2
< -^2 / \g(x + j-nN) - g(j-nN)\2dx\
e nez V"ie	J
( \1/2
x I / \g(x 4- к — nW))2 dx j
1/ rb	\1/2
< - P>2 / 1я(ж + j ~ nN) ~	- nN)\2dx
e \nezJ-h<	J
(	\1/2
x I / \g(x 4- к — n7V)|2 dx j = (♦).
\nez^_4e	J
Via Lemma 10.2.1 the second term in (*) can be estimated by
\ 1/2
|p(x 4- к — n/V)|2 dx ]
thus
\ 1/2
\g(x 4- j - nN) - g(j - nN)\2dx I
which converges to zero for e -> 0 because of condition (R); the proof is completed.	□
In continuation of the above results and with similar techniques, Janssen proves the following result in [188]:
Lemma 10.2.4 Suppose that g 6 L2(K) satisfies condition (R) and that {Em/MTnNg}m,n£Z i>s a Bessel sequence in L2(JR) for some M,N e N. Then any function of the form
Ф = cmnEm/NTnMg, where {cmn} £ fl(Z2) (10.11) 771,nCZ
also satisfies condition (R).
Note that the functions ф in (10.11) are linear combinations of the Gabor system with respect to the dual lattice	Since
the Gaussian g(x) = e~^x satisfies condition (R), see Exercise 9.1, and {E'mTn^}m>nGz is complete in L2(K), Lemma 10.2.4 gives an alternative
10.2 Discrete Gabor systems through sampling 243
argument for condition (R) being satisfied on a dense set of functions in L3(R).
Lemma 10.2.4 has an interesting application to sampling of the frame operator S associated with a frame {Ет/мТпмд}т,пе2 in L2(№). For the proof we will use a result from [187]:
Lemma 10.2.5 Let g € L2(K), M, N E N, and assume that the Gabor system {Em/MTn^g}minEz is a frame for L (IR). If {Em//^Tn^g} satisfies condition (A), then also {Ern/MTn^S~1 g}m nez satisfies condition (A).
Proposition 10.2.6 Let g f= T2(R), M,N € N, and assume that {Em/M^nN9}m,nEZ ™ a Bessel sequence and satisfies condition (A). Then, for any f G L2(R) which satisfies condition (R) and for which {Em/M^nN f}m,nez is a Bessel sequence,
Sf(j) — yy 57 Em/NTnMg}Em/NTnMf(j), j e Z. (10.12) m,nEl
If furthermore g satisfies condition (R) and we denote the frame operator for {Em/MTnN9D}nez,m-o,...,M-i by SD : (2(Z) -> £2(Z), then
(Sf)D = SDfD-,	(10.13)
if we also add the assumption that {Em/MTnNg}m,nE% is a frame, then (S-'gf = (3°Г19°-	(10.14)
Proof. By Theorem 9.2.5, the frame operator has the representation
Sf =	57	Em/NTnwg)Em/NTnMf, f e L2(R).
m,nEZ
If f satisfies condition (R) and {Ет/мТпк f}minEz is a Bessel sequence, then Sf satisfies condition (R) by Lemma 10.2.4. In particular we can sample Sf at the integers; this yields (10.12), with absolute convergence of the series because {{g, Em/NTnMg)}m,nez € ^(Z2). For the proof that the extra assumption implies (10.13) we refer to [189]. Now, assume that is a frame. By Corollary 9.2.4(h),
S g —	5 ? (S g,Em/NTnMS д)Ет^Тпмд-
m,nEEZ
Since {Ет^мТп^З~1 g}minEz satisfies condition (A) by Lemma 10.2.5 and {Ет/^ТпмЗ-1 g}minez is a Bessel sequence, we can apply (10.13) to the function f := S~1g, and (10.14) follows.	□
In words, (10.13) means that we can obtain knowledge about the frame operator for a Gabor system in L2(K) based on the frame operator for a
244	10. Gabor Frames in £2(Z)
Gabor system in £2(Z). Using functional calculus, Janssen has extended (10.13) considerably. In fact, there are conditions such that S can be replaced by <^(S), where 99 is an analytic function. Hereby the obtained sampling results are also applicable to the function S-1/2# which generates the canonical tight frame associated to {Em/MTnN9D}ne%,7n=o,...,M-i. Janssens proof is published in [85].
The canonical dual of a frame {Em/MTnN9D}neZjn==o,...,M-i is given by {Em/MTnN{SD)~1gD}nEz,m=o,...,M-i\ that is, as for Gabor frames in L2(K), it consists of time-frequency shifts of a single function.
10.3	Gabor frames in
In signal and image processing one can only process finite sequences € ^2(Z), and one has to use a finite Gabor system. We shall only give a short description at the end of the section of how the transfer from a model in ^2(Z) to a finite model can be done (see e.g. [265] for more on this aspect), but we will discuss a few aspects of Gabor frames in CL in more detail. Note that Qiu has a series of papers [237], [238] about the structure of Gabor systems in CL.
To connect with the results for frames in ^2(Z) we will write a sequence g G CL as
<; = (#(O),0(1),... ,#(L — 1)).
The definition of the modulation operator on £2(Z) given in (10.1) also defines E^ as an operator on CL. In contrast, the definition (10.2) of the translation operator Tn does not immediately make sense on CL because j — n does not always belong to {0,1,..., L — 1}. The natural way to solve this problem is to extend g G CL to a periodic sequence indexed by Z. That is, we define
g(j + kL) = g(j) for j = 0,..., L - 1, к e Z.
With this convention we can apply the translation operator to sequences in CL.
We prove a result by Qiu und Feichtinger [135] about the Gabor frame operator:
Theorem 10.3.1 Let g € CL and M,N,Nf € N be given. Assume that M, Nr < L. Then the jk-th entry in the matrix representation of the frame operator S : CL CL for {Ет/мТпм9}т=о',п=о ' is 9юеп by
M TnNg(k)TnNg(j) ifj-ke MZ, 0	if j — к MZ.
{Sek,ej} =
10.4 Shift-invariant systems 245
proof. Letting	denote the canonical orthonormal basis for CL,
the j/c-th entry in the matrix representation for S : CL —> CL is
N'-l M-l
(Sek,ej) = E E	Ern/MTnNg} (Ет/мТп^g, ej)
n=0 m=Q
N'-l M-l
=	Ет^мТп^g(k)Em/MEnNg(j)
n—0 m=0
(N'-l	\ /М-l	\
£ TnNg(k)TnNg(j) £ ^ь-Ч/м .
77=0	J \ 771=0	/
Since
( \ л 2тгim(j—к)/м\   ( M j & £ AfZ, "10 if j - к 4 MZt
\m=0	/	1	J	'
this proves the result.	□
In words, Theorem 10.3.1 says that only every M-th sub diagonal in the matrix for S is non-zero. In [135] the result is used to find fast algorithms to calculate the dual frame. Strohmer [266] used Theorem 10.3.1 to obtain factorizations of the frame operator.
Let us conclude with a relationship between frames for ^2(Z) and CL which was observed by Janssen. Given g € ^2(Z) and 7И, N E N such that {Em^MTn^g}nEz,m=o,...,M-i is a Gabor frame for £2(Z), let L be the smallest common multiple of M and V; that is, we can write
L = MM' = NN', where gcd(M',V') = 1.
Assuming that # € ^(Z), we define an element in CL by
Pp0') = E/0 -nL)> J = 0,...,L-l-n£Z
Janssen proved in [188] that {Ет/мТпкдр}М~^=~1 is a frame for CL with the same bounds as {Em/MTnNg}nEz,m=o,...,M-i in £2(Z).
10.4	Shift-invariant systems
As in the L2(K)-case, the discrete Gabor systems in £2(Z) are special cases of general shift-invariant systems of the form
{ffnm (j)}n€Z,77i=0,...,M — 1 — {pTTi(j — П V) }n^z,m=0,...,M —1 j
here each gm is a sequence in ^2(Z). We will always let m,n run through the index set given above, so we will skip the index and simply write {gnm} for the shift-invariant system.
246
10. Gabor Frames in Z?2(Z)
The results for continuous shift-invariant systems in Section 8.7 have discrete counterparts, which are stated in [185]. In order to formulate the results, define the Fourier transform of a sequence h E £2(Z) by
h(v) = ^h(j)e-^iiv, a.e. v € R.
jGZ
Given a shift-invariant system {#nm} we define, analogous to (8.34), the matrix-valued function
= ~ ’ a’e’ y
Observe that this is an N x M matrix.
Theorem 10.4.1 In the setting above the following holds:
(i) {#nm} is a Bessel sequence in ^2(Z) with upper bound В if and only if defines for a.e. v a bounded linear mapping from CM into CN of norm at most VNB.
(ii) {#nm} is a frame for £2(Z) with frame bounds A, В if and only if
NAI <	< NBI, a.e. i/.
(ii) {#nm} is a tight frame for £2(Z) if and only if there is a constant c > 0 such that
M-l
- k/N)gm(v) = c5f.fi, к eZ, a.e. v.
771=0
(iv) Two shift-invariant systems {дПтп} wad {hnm}, which form Beisel sequences in ^2(Z), are dual frames if and only if
M-l	_____
9m^ ~ k/N)hm(y) = N5kfi, kel, a.e. v.
771=0
Most proofs follow by repeating the arguments from the continuous setting, and we will not go into detail. Again, the statements have direct consequences for discrete Gabor frames (Exercise 10.1).
10.5 Frames in £2(Z) and filter banks
Shift-invariant systems appear in signal processing, especially in connection with filter banks. We refer to the book by Vetterli and Kovacevic [282] for a detailed description of this subject and its relationship to signal expansions. We will think about a filter bank as some kind of “black box”, which performs some operations (i.e., processing) on a given input signal and then delivers an output. An example could be that the filter bank performs
10.5 Frames in Z?2(Z) and filter banks
247
an analysis of the signal f via a shift-invariant system {gnm} and then a synthesis via another system {hnm}\ the outcome will be a sequence
f =	(10.15)
m,n
Note that the words “synthesis” and “analysis” correspond to the alternative names of the pre-frame operator T and its adjoint T*, cf. page 90.
The study of the relations between frames and filter banks was initiated by Cvetkovic and Vetterli [102], [103], and has been elaborated on by many other authors (see e.g. [37], [38] [267] and the references given there). We will not go into detail with any of the obtained results, but only guide the reader to the terminology in filter bank theory and provide links to further reading.
The case of a perfect reconstruction filter bank corresponds to {#nm} and {hnm} being dual frames. In this case the outcome f in (10.15) equals /. A paraunitary filter bank corresponds to the special case where {pnm} is a tight frame, implying that we can take hnm to be a multiple of gnTn. A modulated filter bank is a Gabor system, and if it is oversampled , we have an overcomplete Gabor frame.
In the signal processing literature the results are often formulated via the polyphase representation, which is called the discrete Zak transform by mathematicians. For a given signal h € ^2(Z) and a parameter N E N it is defined by
(Z/i)Q>) = ^h(j -lN)e2*ilv, j e Z, a.e. v E R.
zez
An interpretation of the discrete Zak transform is that for a.e. и e К it defines a sequence (Zh)(-, z/). In terms of the polyphase representation one can now define a polyphase matrix, which plays a similar role for discrete Gabor systems as the Zibulski-Zeevi matrix in the continuous case. Among the results in [102], [103] are
•	Characterizations of frames and tight frames in terms of the polyphase matrix.
•	Conditions for the dual frame to consist of vectors in f?2(Z) with finite length (i.e., only finitely many non-zero entries).
•	Characterizations of tight Gabor frames in ^2(Z), generated by a vector with finite length.
Some further results and extensions were later given by Bolcskei, Hlawatsch, and Feichtinger in [38]. Among their results are
248	10. Gabor Frames in £2(Z)
•	A parameterization of all synthesis filter banks providing perfect reconstruction for a given analysis filter bank.
•	Methods for estimation of the frame bounds.
•	Conditions for the shift-invariant system forming the analysis filter bank to be a frame.
•	Construction of paraunitary filter banks from perfect reconstruction filter banks.
Let us finally mention the paper [267] by Strohmer, where he provides methods for approximation of the canonical dual frame associated to a shift-invariant frame.
10.6 Exercises
10.1	Derive the consequences of Theorem 10.4.1 for discrete Gabor systems.
10.2	Here we ask the reader to prove an extension of Proposition 1.3.3. In fact, show that for a bi-infinite matrix A = {Am,n}m nGZ, the following are equivalent:
(i)	There exist constants A, В > 0 such that
Ic^|2 - HMck}||2 < в£Ы2 for all finite sequences {cfc}.
(ii)	The columns in A constitute a Riesz basis for their closed span in £2(Z).
(iii)	The rows in A constitute a frame for £2(Z).	?
11
General Wavelet Frames
The fundamental question in wavelet analysis is what conditions we have to impose on a function such that a given signal f E L2(R) can be expanded via translated and scaled versions of ip, i.e., via functions
Г'Ь(х) := (Дади = rrUV’l—), a / 0, b 6 R. (11.1) lai1/2 a
Thus there is a basic similarity between wavelet analysis and Gabor analysis: both concern sequences of functions defined by letting a special class of operators act on a fixed function, i.e., in both cases we are dealing with coherent systems. The connections are even closer, and both can be seen as manifestations of a theory, to which we return in Chapter 17. As in Gabor analysis there are two ways in which one can think about expansions of a signal f in terms of the functions 'фа'ь. One way is to ask for representations of f as integrals involving wa,b over tf2. Alternatively, one can restrict the parameters a, b to a discrete subset A of Ж2 and ask for series expansions of f in terms of the corresponding functions 'фа,ь. For applications the latter is usually the most convenient choice, and most of this chapter will deal with the question of how we can choose the discrete subset A and such that {'0а,&}(а,ь)ел is a frame for L2(JR).
Collections of functions of the type (11.1) have been used in different contexts a long time before the wavelet era began, see for example the construction by Haar discussed in Example 3.8.1. Morlet was the first to propose representing signals as integrals involving 'фа,ь, and together with Grossmann he introduced in [149] what is now known as the wavelet transform. Again it was Grossmann who brought the theory a big step forward
250
11. General Wavelet Frames
by proposing to construct frames consisting of a countable number of functions 'фа'ь. Together with Daubechies and Meyer he published the first constructions in [108].
Another breakthrough came with the concept of multiresolution analysis, as developed by Mallat and Meyer. As discussed in Section 3.8 its original purpose is to construct orthonormal bases for L2(IR) of the form - k}}i<k&z. The importance of this new subject was immediately recognized by the mathematical as well as the engineering community and very soon most of the effort in wavelet analysis was concentrated on construction of orthonormal bases with prescribed properties. Nowadays “wavelet analysis” is for many people almost synonymous with “multiresolution analysis”, but wavelet analysis is in fact a much broader subject. For historical accuracy the reader is encouraged to consult the paper [105] by Daubechies, which was written around the time when multiresolution analysis was introduced. The paper contains a large number of important wavelet results, but multiresolution analysis is barely mentioned. The same remark applies to the excellent survey paper [172] by Heil and Walnut, which was published in 1989. At that time one could certainly not predict that soon almost all effort would go into constructions via multiresolution analysis.
This chapter and the following three chapters will deal with different aspects related to overcompleteness of collections of functions of the form (11.1). As discussed in Section 4.2, overcompleteness is introduced in order to obtain more flexibility and be able to make constructions which can not be done with e.g., orthonormal bases. We follow the historical development and begin by constructing frames without any multiresolution structure. Frames based on various multiresolution schemes are discussed in Chapters 13-14.
A few words on terminology are needed. The word wavelet is usually reserved for a function for which
{2^(Vx - fc)h,fcez = {ф2~’’2~‘к}1,кег	(П.2)
is an orthonormal basis for L2(IR). We will follow this tradition, but the word “wavelet” will appear in several constellations. Since we are interested in more general ways of choosing the translates and dilates than in (11.2), we will call any discrete family of the type {4>a,b}(a,b)eM Л C IR2, a wavelet system. A family of functions which consist of translated and dilated versions of a single function, is said to have wavelet structure.
We begin with a short section on the continuous wavelet transform, which delivers integral representations of each f E L2(R) of the type
f=[ f cf(a,b)^a’bdadb,	(11.3)
J —oo «/ —oo
provided that satisfies some admissibility conditions and that the integral is interpreted in the right sense.
11.1 The continuous wavelet transform
251
Then we move to construction of frames for L2(K) consisting of functions of the type {^а,&}(а,ь)ел- The obtained representations can be considered as discrete versions of (11.3), but our presentation does not rely on any result about the continuous wavelet transform. We begin in Section 11.2 with the (regular) case, where the dilation parameter “two” in (11.2) is replaced by a number a > 1 and integer translation is replaced by translation with a step size b > 0, i.e., we consider wavelet systems of the form {a^2^{a}x — kb)}jtke%. In Section 11.3 we discuss certain irregular choices of the discretization.
11.1 The continuous wavelet transform
Let ф € L2(R). We say that ф satisfies the admissibility condition if
^:=Г^7<оо.	(11.4)
J— oo |7|
We also say that is admissible. Note that if $ is continuous in 0, which is e.g., the case if G LX(R), then (11.4) can only be satisfied if -0(0) = 0, i.e., J-oo i/>(x)dx = 0. But if this condition is satisfied, relatively weak decay conditions on $ imply that (11.4) is satisfied.
Given an admissible function -0 € L2(JR) we define the continuous wavelet transform with respect to <ф of the function f e L2(K) as the function W^(/) of two variables given by
Proposition 11.1.1 Assume that is admissible. Then, for all functions f,9 e L2(R),
/	/	W^f)(a,b)W^g)(a,b)^=C^f,g}.	(11.5)
«/ —OO J —oo
Proof. Using the commutator relations for the Fourier transform and the operators Tb,Da, (/)(«,&) = (/,^ь)
= (Е/,ЕТьОаф)
= (j, Е_ьП1/а'ф')
252
11. General Wavelet Frames
If we for a moment consider a fixed value for a, this expression is the Fourier transform of the function
^a(7) = /(7)|a|1/2V'(«7)I
calculated in the point —b. If we define Ga(ff) similarly, it follows that
f W^f)(a,b)Wi,(g)(a,b)db = [ Fa(-b)Ga(-b)db
J —oo	J —oo
= (Fa,Ga)
= <Fa,Ga)
= [ fb')9h) lal 1^(«7)|2^7-
J —oo
Inserting this expression in the left-hand side of (11.5) and using Fubini’s theorem gives
Г oo Г OO	_____________J J L
/	/	W^f){a,b)W^a,b) —
J-oqJ-oo	a
Zoo fOO л ________ л
/ /(7)5(7) H |-0(a7)|2<^7—2-
-00 J — 00	a
= [ ([ Дйа7)|2<^) /(7)р(7)<^7-
J— 00 \J — 00 1^1	/
By a change of variable,
[ ДНа7)|2^ = [ Д|Иа)|2<*а = С*;
J-OO H	J-00 PI
thus
Г Г Wi,(f)(.a,b)Wi,(g)(a,b)^- = C^f,g)
J-oo J-QQ	О
= c^f.gY
□
As in the Gabor case we write the result as
/°° Г 00	dndh
/ ^(/)(а,бЖ-«,/eb2(R),	(u-б)
-00 J —00
where the integral is understood in the weak sense. We refer to Proposition 17.1.6 for a similar statement where pointwise convergence is obtained via stronger conditions.
Corollary 11.1.2 If'ipe L2(R) is admissible, then {^а,&}а^о,Ьек is a continuous frame for L2(R) with respect to RxR\{0} equipped with the measure Kdadb.
11.2 Sufficient and necessary conditions 253
Ц.2 Sufficient and necessary conditions
^Ve now turn to the construction of (discrete) frames having the wavelet structure. We will first consider the case where the points (a, b) in (11.1) are restricted to discrete sets of the type {(a-7, kba^)}^^, where a > 1, b > 0; a is the dilation parameter or scaling parameter and b is the translation parameter. We hereby obtain the functions
(Tkba,Da,^)(x) = (ра1Ткьф)(х) =	г-
Re-indexing (i.e., replacing j by - j) we see that
= {а^2-ф(а>х - kb)}3,ksZ.	(11.7)
Definition 11.2.1 Let a> 1,6 > 0 and ^eL2(R). A frame for L2(R) of the form {а^2ф(^азх — kb)}j^z is called a wavelet frame.
The main purpose of this chapter is to present sufficient conditions for {а^2ф{адх — A;6)}^kGz to be a frame. The results will be stated in terms of the functions
Go(7) = £ |^(^7)|2,Gi(7) = E E |W(<6 + ВД1.7 € R- (И-8)
Because we usually consider fixed values of a, 6, the dependence of these parameters is suppressed in the notation. Note that
G0(a7) = G0(7), G^a^G^).
Geometrically this means that the graphs for Go,Gu for |7| e [a-7, a-74"1] are stretched versions of their graphs for |7| e [a7-1, a7]. See Figure 11.1 for an illustration based on the Mexican hat wavelet described in Example A.8.2. It follows that
supGfc(7) = sup Gfc(7), infGfc(7)= inf Gfc(7), к = 0,1.
7GR	|7|6[l,a]	7GK	|7|e[l,a]
The role played by the functions Go and G^ in wavelet analysis corresponds to the role of the functions G and |-Hfc(z)| defined in (8.10) and (8.25) in Gabor analysis. Similar to Proposition 8.3.2, but technically slightly more involved, the following necessary condition for {ai^aix _ £5)} j,jfeez to be a frame was proved by Chui and Shi [95].
Proposition 11.2.2 Let a > 1,6 > 0 and ф e L2(R) be given. If {ад/2ф(адх — kb)} j ,kez is a frame with frame bounds A,B, then
ЬА <	|'0(aJ7)| < bB, a.e.
j€Z
254
11. General Wavelet Frames
Figure 11.1. The functions Go (the upper graph) and Gi based on the Mexican hat and the parameters a = 2, b = 1.5.
Theorem 11.2.3 Let a > 1,6 > 0 and if e L2(R) be given. Suppose that
В := | sup |^(aJ7)^(aJ7 + k/6)| < oo.	(11.9)
Then {а^2,ф(а? x — kby^kez is a Bessel sequence with bound B, and for all functions f e L2(R) for which f e Cc(R),
E \{f,DaiTkb^\2 = | Г |/(7)|2Е|Ж7)|2<*7	(U-Ю)
'/-o°	JGZ
+zEE [ f(y)f(y - aik/b^a-iy^a^y - k/b)dy.
0 k^o jez J"00
If furthermore
E |^(aJ7)| “EE	+ fc/6)|
jGZ	fc0OjGZ
(11-11)
then {а^2гф(а^х —	is a frame for L2(R) with bounds A, B.
11.2 Sufficient and necessary conditions
255
proof. Let f 6 L2(JR) and assume that f is continuous and has compact support. We begin with some calculations leading to (11.14) below, which will be a key ingredient in the proof. Fix j 6 Z. Then
— а3к/Ь\ф(а 7
__ f0-3 lb л
52 / l№ “ a3k/b)^a~3^ ~ k/b)\dy kezjQ

|/(7)Vi(a •J7)|d7
1/2
OO.
Thus we can define a function Fj : JR. -> C by
Fj(y) ~ 52	a3k/b)$(a-^ - k/b), a.e. y.
к GZ
Fj is aJ/6-periodic, and the above argument gives that Fj G L1(0, a3/b). In fact, we even have Fj 6 L2(0,aJ/b). To see this, we first note that
|Fj(7)|2 < £ 1/(7 - E 1^7 - W-fcez	fcez
Since f e Cc(lR), the function 7 ->	1/(7 — a^k/b)\2 is bounded; now
an argument similar to above shows that Fj e L2 (0,aJ/b). We leave it to the reader to verify this, and also that
[	= Г'b	(11.12)
«/ — 00	Jo
Since {a~3^2b1^2Ema-3b}m ez is an orthonormal basis for L2(0,aJ/5), Parseval’s equality shows that
E /	^(7)e2%ima ^7
meZ
a-7 fa3/b
= ~ь]о IWI2<*7;
(11.13)
combining (11.12), (11.13) and the definition of Fj, we obtain that
256
11. General Wavelet Frames
E Г O(a-’7)e^4
mEZ
(11.14)
a> Га,/Ь
~b Jo
2
/(7 — aJ к/ЬУф^а-з^ — k/b) dy. к GZ
We now prove (11.10) for our special choice of the function f. The most delicate point in the proof is several interchanges of sums and integrals. In order to make the argument rigorous we will first show that КЛ ^a^kb^)\2 is finite by replacing all occurring functions by their absolute values. For positive functions all interchanges are allowed. After showing that the sum is finite, all calculations can be repeated without absolute sign to get the exact expression (11.10). We will not perform the repetition, and for that reason we keep all occurring complex conjugations in the first part of the calculation, even though they are superfluous for the first part.
The first step is to use the commutator relations for the Fourier transform and the operators Da,Tb'.
E к/,^ад|2 = E E itf-
Mez	jezmez
= E E \{^f^DaiTmb^\2 jezmez
=	^)i2
j’GZ m(zZ
jeZ mEZ
?oo ________ .	2
= E^E /
jeZ mEZ
= (*)•
Since we are summing over all j 6 Z, we can replace j by — j; doing so,: and continuing using (11.14), we have
•a3/b
___ „j ra /b
jez J0 kez
| ra3 /b ______л	Z
lE /	-kjb)\ dy
jez J °	kez
— a/k/b)tp(a~iy — k/b) dy
2
2
(ll.!^
11.2 Sufficient and necessary conditions 257
Using that |c|2 = cc for any complex number,
(**)
1 rajlb (	_____________
< ь E / E l№ - аЧ/b^a-h - e/b) |
0 jezjQ \eez
x |/(7 — аЭк/ЬУф^аГ^ — k/b)\ j dy kez	/
1	с0,3 /ь ,	-------------
= -b ЕЕ/	(i/( 7 — a-7^/5)'0(a_^7 — £/b)\
° jezeez Jo
x У2 1/(7 — & к/ЬУф(а~17 — /с/Ь)Н dy fcGZ	/
= (***).
The function 7 -»	|/(7 — aik/ty^a — k/b)\ is c^/fr-periodic, so
we can continue with
r E / l/(7)V’(a-J7)l  52 1Л7 - аЩЬУф^а - fc/b)M7 jez’7-00	fcez
T 52 E [ l/(7)/(7 - a>fc/&)^(a-J7)V>(a-J7 - k/b)\dy
° kezjezJ~°°
T [ l/(7)|2 52l^(^7)|2d7
J~co
+ т E E [ |/(7)/(7 - а^/&)^(а-^7)^(а_7 - fc/b)| dy
Т Г |Л<-ЕЙа~7)|2<Ъ+|я, J~°°	jez	0
-R = 52 E [ l/(7)/(7 - a-’fc/b)^(a--’7)’/'(a-J7 “ VW7-
k#0 JGZ
where
We now estimate the term R. Using Cauchy-Schwarz’ inequality twice, first on the integral and then on the sum over k, we obtain
258
11. General Wavelet Frames
/ Г°° ~	л	\ 1/2
R < 52 52 ( /	|-0(a_J-y) Tpla-^- k/b)\dy \
je^k^O V-oo	/
/ roc	\ 1/2
x ( /	|/(? - a3k/b}\2 \ф{а~3^} ,ф(а~3^ - k/b)\dr(\
\J —00	/
/00	\ 1/2
< 52 ( 52 [ l/WI2 lV’(a“J7) ^(.aT^ -k/b)\dy j jez yk/o“—00	)
(<x	\1/2
x I ^2 [ 1^7 “ aW6)|2 |V’(a“J7) ’Ф(а~3у - k/b)\dy j
7/o7~°°	/
= 52w(**)’ jgz
where
/00	\ 1/2
(*)	= |52 [ l/(7)|2 |^(л-7) ^(a_J7-fc/b)l<*7 I ,
7/o J/
/ eo	\ 1/2
(**	) = [52 [ |/(7 ~ a7Vb)|2 l^(e-,7) ^(a~j7- k/b)\d-y )	.
7/o J-°°	/
The terms (*) and (**) are actually identical; in fact, by the change of variable 7 -> 7	4- aik/b in (**),
/00	\	1/2
(**)	=	[ 52/ l/(7)|2 |^(a-37 +fc/Ь) 7a~J7)H7]
7/o7-00	J
/00	\	1/2
= (52/' l/(7)|2 l^(a~J7W(a“J7~ V&)ld7 )
= (*)•
Therefore,
R < 52 E [ |/(7)|2 l^(a-J7) 7a-,7-fc/b)l<h>
11.2 Sufficient and necessary conditions
259
and it follows that
E \{f,DalTkb^\2 j,kez
< v [ l/(7)|2 E
° V-oo	JSZ
+| [ |/Ъ)|2ЕЕйа-^(а-^-*/ь)1<*у
k^ojez
= 7 [ \fh'l\2^/^\^(,a~:'y^a.-3'r-k/b)\d'r.
kezjez
Note that
E 52 |V’(“-'’7)^(«-J7 - k/b)\ = E E l^(aJ7M®,7 + k/b)\;
kezjez	kezjez
using the assumption (11.9) we therefore have
E i(/.^w)i2 < в н/п2
j ,k£Z
= В ll/ll2.
Since this holds for all functions / for which / is continuous and has compact support, it holds for all / 6i2(K) by Lemma 3.2.6; thus the wavelet system {а^2)ф(а^х — kb)}jtkez is a Bessel sequence with bound B. We can now go back and repeat all the above calculations without absolute sign, still with / e Cc(R), to get the announced expression (11.10) for Sj kez I(/’ DaiTkb^W since we can just skip the single inequality (11.15) we obtain an exact expression. If we also assume that (11.11) is satisfied, then
E \{f, DaiTkb^}\2
j,kez
T [ l/(7)|2 ^bkahyi2
J~°°	jGZ
- ?EE / /(?)/(7 - а3к/Ь)-ф(а~^)'ф{а 3y - k/b^dy fc?SO ]&J-CO
у + k/b)\ dy
> л ц/n2.
The proof is now completed via Lemma 5.1.7.
□
260
11. General Wavelet Frames
If $ is continuous in 0, which is e.g., the case if e L1(IR), then the condition (11.9) can only be satisfied if ^(0) = 0, because -> -0(0) as j —> — oo. If this necessary condition is satisfied, then very reasonable conditions on -0 will imply that {aJ^2u^aJx —	is a frame whenever
b is sufficiently small. We need some lemmas before we prove a formal version of this statement in Proposition 11.2.6.
Lemma 11.2.4 Let x,y G JR. Then, for all 8 e [0,1],
1
1 4- (x 4- y)2
1 4- x2 \
1 + У2 )
( 2\5
Proof. Given x,y 6 IR, the function 8 -> 2 (is monotone, so it is enough to prove the result for 8 = 0 and 8 = 1. The case 5 = 0 is clear; for 5 = 1, we use that 2ab < a2 + b2 for all a, b 6 IR to obtain that
1+y2 = 1 + ((j/ + x) - x)2
= 1 + (y + x)2 + x2 - 2x(y 4- x)
< 14-2(0/ + ж))2+ж2)
< 2(1 + (у + ж)2)(1 + x2).
Lemma 11.2.5 Let <ф e L2(IR) and assume that there exists a constant C > 0 such that
l^(7)l < £(! + |7|2)3/2 a-e-
Then, for all a > 1 and b > 0,
12 ZL l’/'(«J7)V’(aJ7 + fc/6)|
/ n2
16C2&4/3 (	+
\ a — 1
a
a2/2 - 1
(11.16)
Proof. The decay condition on gives that
|'0(c?7)'0(a-77 4- k/b)\
\ajy\	|aJ7 4- k/b\
(1 4-|^7|2)3/2 (1 4-|^7 4-V^|2)3/2
\aJ7|	(1 4- |nJ7 4- A;/&|2)1//2
(1 4- |aJ7I2)3/2 (1 4- |a>7 + A;/&|2)3/2
|aJ7l_______________1_________
(1 4- |a->7|2)3/2 1 4- |a->7 4- A;/&|2 ‘
11.2 Sufficient and necessary conditions 261
Applying Lemma 11.2.4 on (1 + |aJ7 + k/b\2) 1 with 5 = | gives
l^'>'W7 + W < 2C,2(1 + ^2)3/2 (i +|fc/&|2)
< lfl^l ( 1 У7"
~	(1 + |aJ7|2)5/6 \1 + |fc/6|2/
In this last estimate, j and к appear in separate terms. Thus,
52 52 IV’(aJ7)’/'(aJ7 + k/b)\
2C2 (V' la37l fv' (	1
l^(l + l^7|2)5'6 ^V + IW/ / \ JtZAJ	/ \Avy-U	/
(H-17)
For the sum over к / 0,
&V3 (62 + fc2)2/3
k=l
< 2b4/3 (J™ t-4/3dt+l
= Sb4'3.
In order to estimate the sum over j e Z in (11.17) we define the function
^) = E;l4.!aX/6.7 6R.
(1 + |^7|2)5/6
We want to show that f is bounded. Note that /(ay) = f(y) for all 7; it is therefore enough to consider |y| G [1,a], so we can use that
|aJy| < aj+1, 1 + |aJy|2 > 1 + a2-7.
262
11. General Wavelet Frames
Thus
y- |qJ7l
“ (1 + |aJ7l2)5/6
a3T ^(1+«2J)5/6 J
0	74-1	°
EaJ±i 5-j=_oo (1 + a2>)5/6 + £
0	oo 7-_|_ 1
aj+1
(l + a2>)5/6
1
a~2/3
1 — a-2/3
a
ai-a-1
a — 1 + a2/3 — 1
That is, f is bounded as claimed. Putting all information together, and using (11.17),
52 52 IV’(aJ7)^(a-’7 + k/b)\
k^OjGZ
<	2C2 I V	i
\irt t1 + laJ7|2)5/6 I \J t *-*	/
/	2
<	16^/3	“	+	«
\ a — 1	a2/3 — 1
1 \2/3
1 + IW7
□
We are now ready to give sufficient conditions for {aJ/2^(aJ\r — kb)}j^Ez to be a frame for small values of b:
Proposition 11.2.6 Let ф e L2(R) and a > 1 be given. Assume that
(i) inf|7|e[i,o]£JeZ|Vi(a-'7)|2 >0.
(ii) There exists a constant C > 0 such that
\^)\<c	/2 a.e.	(11.18)
(1 4- |7I ) 7
Then {a-7/2'0(a-7\r — kb)}j^i is a frame for L2(K) for all sufficiently small translation parameters b > 0.
Proof. We first prove that {a^^^ajx — kb)}j^i is a Bessel sequence for
all b > 0. Arguments similar to the one used in the proof of Lemma 11.2.5
11.2 Sufficient and necessary conditions
263
shows that (Exercise 11.1)
+	(11.19)
jez
Via Lemma 11.2.5 it follows that
EE |V>(aJ7)V>(aJ7 4- k/b)\ ke%je% (2	\	i	4
+ ^7Г—г) + -Г-Г + -r-r;
a — 1	a2'3 — 1 / a4 — 1 a2 — 1
by Theorem 11.2.3, we conclude that {а^2ф(а^х - kb)}jyke% is a Bessel sequence. By choosing b sufficiently small, the assumption (i) implies that
(/2	\ \
-I6C4V1 + > °- <u-20>
JGZ	4	7 J
and in this case, by Lemma 11.2.5,
»nf ( EI^t'^'EEI^tV^t + V6)! j >o.
17161 \з&г	k^o jez	/
Theorem 11.2.3 now gives the desired conclusion.	□
The proof of Proposition 11.2.6 shows that {а^2ф(а?х — kb)}j,kE% is a frame whenever b > 0 satisfies (11.20). In concrete cases we can often use much larger values of b:
Example 11.2.7 Let a = 2 and consider the function
ф(х) = -^=7Г-1/4(1 - x2)e~^x .
Due to its shape, ф is called the Mexican hat. As proved in Example A.8.2, ^(7) = 8^/|7Г9/472е“2’г2'1'2.
A numerical calculation shows that
inf V |^(2^7)|2 > 3.27.
Also, (11.18) is satisfied for C = 4, so a direct calculation using (11.20) shows that {2J/2'0(2J(r - kb)}3tke% is a frame if b < 0.0084. This is far from being optimal: numerical calculations based on the expressions for A, В in Theorem 11.2.3 gives that {2^2ф(23x - kb)}is a frame if b < 1.97! The obtained frame bounds A, В for some selected values for b are as follows:
264
11. General Wavelet Frames
Figure 11.2 The functions in (11.21) for 7 € [|,4], b € [1,1.6]. The upper graph is the function (7, b) i-> ^Оо(у)- Observe the wave-shape in the first coordinate 7 and that the surfaces are getting closer for large values of b.
b	0.25	0.5	0.75	1	1.25	1.5	1.75	1.97
A	13.1	6.55	4.36	3.26	2.33	1.25	0.422	0.0069
В	14.2	7.1	4.73	3.57	3.09	3.13	3.5	3.54
For small values of 6, the frame bounds are almost identical to the values obtained in [106] via a different criterion. For large values of b the bounds above are sharper (for b = 1.5, the bounds given in [106] are A = 0.325 and В = 4.221). Furthermore, the criterion used in [106] suggests that the frame property breaks down already before b = 1.75.
We can illustrate the dependence on b by a 3D-plot of the functions
(7,6)^1Go(7),	(11.21)
b	b
where Gq,Gi are defined in (11.8). In terms of these functions, Theorem 11.2.3 says that {2^2'ф(2^х — kb)} j,kez is a frame with lower frame bound
This page is lost
266
11. General Wavelet Frames
A if
A:= inf 1(Go(7)-G1(7))>0.	(11.22)
7б[1,2] b
As upper frame bound we can use
B= sup 1(Go(7)+Gi(7)).
76(1,2] 0
Figure 11.2 shows the graphs of the functions in (11.21) for variables 7 € [|, 4], b e [1,1.6]. There is a positive minimal distance between the surfaces, so it is clear that (11.22) is satisfied for all b € [1,1.6]. Note also that the “gap” between the surfaces is increasing when b is getting smaller and that |Gi(7) -» 0 uniformly for b —> 0. This implies that the ratio of the frame bounds for {2J/2'0(2J\r — kb)}jfkez is approaching 1, i.e., that the frame is closer to being tight, the smaller b is.
For b e [1.6,1.8], Figure 11.3 shows that {2^2/ф(2^х — kb)}j^ez is still a frame, but the gap between the surfaces is small for b = 1.8. The frame is far from being tight in this case. Finally, Figure 11.4 shows that the criterion (11.22) is no longer satisfied for b ~ 2.	□
The Fourier transform of the Mexican hat decays much faster than assumed in (11.18), so it is not a surprise that direct estimates via Theorem 11.2.3 gives that {2^2'ф(2^х — кЬ)} is a frame for larger values of b than suggested by Proposition 11.2.6. The same will happen for all functions for which *ф decay faster than assumed in (11.18). In fact, it is the decay of that will make |'0(aJ7)'0(aJ7 + k/b)\ small, so when decays much faster than (11.18) it is clear that	l^(aJ7)^(aJ7 + V&)l will
be significantly smaller than the bound in (11.16). Even for the function ф given by
the estimate in Proposition 11.2.6 is far from being sharp: a numerical estimate shows that
inf УММ2 - 1.5, Ned^]^1
which implies that {2J/2'0(2J\r — kb)}j,kez is a frame if b < 0.037, while numerical estimates based on Theorem 11.2.3 gives that b < 0.24 is sufficient (Exercise 11.2). Proposition 11.2.6 is mainly interesting because it gives the existence of an interval ]0, bo[ such that all translation parameters b belonging to the interval leads to a frame; it should not be used to estimate b0.
There is one remarkable difference between Theorem 11.2.3 and Theorem 8.4.4 for Gabor frames: in the condition for the lower bound in the Gabor
11.3 Irregular wavelet frames
267
version, it is the sum over к of
g(x — na)g(x — na — k/b)
that has to be subtracted from Z^zl#^ “ nu)|2. That is, the absolute sign is outside the sum over n. This is in contrast to the condition in Theorem 11.2.3, where the absolute sign is inside the sums. The condition in the Gabor version is clearly the best, since the position of the absolute sign opens up for possible cancellations. For a = 2, it is known that the condition in Theorem 11.2.3 can be replaced with a condition where the absolute sign is outside, cf. [105], Theorem 2.9. It would be interesting if Theorem 11.2.3 above could be generalized that way.
A sufficient condition for {a^2(ip(a^ x- kb)}jtkez being a Bessel sequence was obtained by Chui and Shi in [91]:
Proposition 11.2.8 Let 3 : [0, oo[-> [0, oof be a function which is nondecreasing on [0, non-increasing on [^7, oof, and for which
[ 0(?)(i + -)<h < oo.
Jo	1
Then every function ip G L2(R) for which
IV’(7)l < ^(l?)l a.e.
generates a Bessel sequence {a?l2(i[>(ajx — kb)}jtkez for any b > 0,a > 1.
In [91] it is also proved that under very reasonable conditions, {a^txplaj x — kb)} j,ke^ being a Bessel sequence is equivalent to the function 7 i->	l'0(a^7)|2 being bounded, which again is equivalent to
/Xo — 0- We refer to the original paper for the exact conditions for these equivalences to hold.
11.3 Irregular wavelet frames
In Section 11.2 we exclusively considered translations with integer-multiples of the parameter b and dilations by j € Z. A more general and considerably more complicated question is:
Which conditions on a discrete sequence {(Aj,in JR+ xR and a function ip 6 L2(JR) imply that
{Xy2ip(XjX - Hj)}jei is a frame for L2(JR)?
A frame of this type is called an irregular wavelet frame. Only few results about irregular wavelet frames are known (see Theorem 11.3.4) and
268
11. General Wavelet Frames
e.g., the proof of Theorem 11.2.3 does not extend to general sequences {(Aj,	But if we assume that the translates are still of the type 6Z,
the essence of Theorem 11.2.3 carries over. There are a few points where extra caution is needed, especially because {A^J-j^z is usually different from {A”1 }jez; in the proof of Theorem 11.2.3 we were frequently switching between {ai}jEz and {a~3}jEz- Also, the function 7 i->	1'0(7/%)P is in
general not periodic, so in order to find its supremum or infimum we have to investigate all 7 G JR. We encourage the reader to check that the proof of Theorem 11.2.3 works in the irregular case with these modifications taken into account (Exercise 11.3). A direct proof is in [78].
Theorem 11.3.1 Let {Aj}jgz be a sequence of positive real numbers, b > 0 and G L2(IR). Suppose that
A := s “ fe ЙГ>I2 ~ E E	> “•
3 k^ojez j j /
and
B := Isup 52	+ t)I < °0-
Then {X1J/2^(Xjx - kb)} j,kez is a frame with frame bounds A and B.
One can check that parts of Chui and Shi’s proof of Proposition 11.2.2 works for irregular wavelet frames of the type {Ay2V,(Aj7 —More precisely:
Lemma 11.3.2 Let G L2(IR). If {Aj}jgz is a sequence in IR-1" and {Aj//2'0(Aj7 - kb)}jtkEz is a frame with upper bound В for some b > 0, then
-B’ a-e-?-
b A>
Lemma 11.3.2 puts restrictions on the sequences {Aj} for which {xy^(xn-kb)} can be a frame. Let us be more specific about this point. Following [82], we say that a sequence {Aj}jGz of positive numbers is logarithmically separated by A > 1 if
I log Aj — log Afc| > log A, Vk/j.
If {Aj}jGz is ordered increasingly, this is equivalent to > A, Vj G Z.
Proposition 11.3.3 Let G L2(JR) and {Aj}jGz be a sequence in IR+. Suppose that {Х^'ф^у-кЬ)} j,kez is a frame and that ф is continuous at a point 70 where ^(70) / 0. Then {Aj}j€z is a finite union of logarithmically separated sets.
11.3 Irregular wavelet frames 269
proof. Let Sj
y-. According to Lemma 11.3.2,
< В, a.e. 7.
jGZ
(11.24)
We can assume that 70 > 0. Let с := |V>(7o)12 and choose J > 0 such that for all 7 E Io ’= [70,7o 4- £[,
|-0(7)|2 > I-
By taking 7 = 1 in (11.24), we see that the number N of elements from {sjjjez belonging to the interval Zo satisfies < B, i.e., N < ^-b. Now let a := and define the intervals 70
Ik '= [7o<A 7ocr':+1 [•
Clearly	is a disjoint covering of B+ (IR+ = UkL_ooIk), and for
given к E Z, the interval Ik contains at most N points from {sj}jEz. Now observe that each point in Io is logarithmically separated from points in the intervals /±2,Л-4, • • • • Similarly, a point from Ц is logarithmically separated with points from the intervals /-1,/±3,/±5,.... Thus {sj}jez can be split into at most 2Ar logarithmically separated subsequences, from which the result follows.	□
Proposition 11.3.3 excludes the frame property for many types of sequences {Ay}j6Z’ If for example Aj = ja for some a > 0 and for j larger than a certain J > 0, then {Х^гр^у - kb)}j,kE% can not even be a Bessel sequence if <ф satisfies the very weak condition in Proposition 11.3.3.
Recently, Sun and Zhou obtained one of the first useful results concerning wavelet frames where both the dilation and the translation are allowed to be irregular:
Theorem 11.3.4 Let E L2(JR) be a real-valued function for which all the functions
x —> ж'0(ж), x -> ^'(z), x
are in L2(R). Assume that -0(0) = 0. Then there exist constants a > 1,5 > 0 such that
J	\ c • i /I
\	/ J j,ke%
is a frame for L2(R) for all sequences {(sj^, p>j,k)}jykEZ for which
(sj,k, C [aJ, aj+1] x [a^bk, aib(k 4- 1)], j,/cEZ. (11.25)
Theorem 11.3.4 can naturally be considered as a perturbation result; in fact, the conditions imply that	jjj^gz is a
270
11. General Wavelet Frames
frame, and the condition (11.25) is “strong enough to guarantee that fc )b,feez is so close t0 la~j/2'll}^x~^Tbk-)}j,kez that it is itself a frame”. We leave this as a rather intuitive statement, but we return to general perturbation theoretic methods in Chapter 15. For the full proof of Theorem 11.3.4 we refer to [274].
11.4 Oversampling of wavelet frames
If	— fcb)}j,kez is a frame for L2(R), the general frame theory tells
us that the wavelet system contains enough elements to represent arbitrary functions in L2(]R) as infinite linear combinations of the frame elements. It is clear from the definition of a frame that a wavelet system Ф containing a frame {а^2>ф(а^х — kb)}jykEz is itself a frame if and only if Ф is a Bessel sequence. An example of a wavelet system containing {а^2ф(а^ x—kb)}jikEz is
{а^2ф(а^х - kb/n)}jikEz,	(11.26)
where n € N. We say that the wavelet system in (11.26) is obtained via oversampling of {a^fox - kb)}jtkEz-
Chui and Shi investigated the frame properties of an oversampled wavelet system in [91]:
Proposition 11.4.1 Assume that {а^2ф(а?х — kb)}j,kEz is a wavelet frame, and that ф satisfies the conditions in Proposition 11.2.8. Then the wavelet system in (11.26) is a wavelet frame for any n G N.
Oversampling will in general change the frame bounds, and for a tight wavelet frame it might happen that the oversampled frame is no longer tight. A positive result was obtained in [91], where the given conditions imply that {(р/2ф[сРх — kb/n)}jtkEz is tight if {а^2ф(а^х — kb)}j>kEz is tight:
Theorem 11.4.2 Let a > 2 be a positive integer and b > 0. Suppose that {а?/2ф((Рх — kb)}jikEz is a frame for L2(JR) with bounds A,B. Then, for any positive integer n which is relatively prime to a, the family in (11.26) is a frame for L2(]R) with bounds nA,nB.
In the special case a = 2, we see that tightness is preserved if n is odd. There exists examples, showing that tightness might not be preserved if n is even, cf. [93].
In this chapter we have concentrated on obtaining frame properties for a wavelet system {аМ2>ф((Рх — kb)}3fkEz] no question about properties of the resulting frame has been investigated yet. We will come back to this subject in Chapters 12-14; in particular, we will see in Example 12.1.1 that
11.5 Exercises
271
the canonical dual of a wavelet frame might not have the wavelet structure. However, there are cases where one can find another dual, which is also a wavelet system, see Theorem 12.1.3. Z/the frame {а^2'ф(а:’х — kb)}j,kEz has a wavelet dual {а^2,ф(а?х - kb)}jtkE% and n is a positive integer which is relatively prime to a, then the oversampled system (11.26) also has a dual with the wavelet structure, namely {^a^2ip(a^x — kb/n)}jykez- We refer to [91] for a proof.
We note that an alternative approach to oversampling was given by Ron and Shen in [247].
11.5 Exercises
11.1	Prove the inequality (11.19).
11.2	Consider the function given by (11.23) and find, based on Proposition 11.2.6, respectively Theorem 11.2.3, values for &o such that {2J’/2'0(2-7\r - k)}jikEz is a frame for all b e]0, b0].
11.3	Prove Theorem 11.3.1.
12
Dyadic Wavelet Frames
In this chapter we consider dyadic wavelet frames, i.e., frames for L2(R) of the type {2J/2'0(2J\r —	Bases of this type were considered already
in Section 3.8, where we also introduced the short notation {ipj,k}j,ke% and {D^Tk'ipjj^ke^- A frame of this type is said to be generated by W
Although dyadic wavelet frames just correspond to wavelet frames with dilation parameter a = 2 and translation parameter b = 1, this chapter is independent of Chapter 11. We will discuss topics which have only been considered in the dyadic case. Sometimes this has technical reasons, but sometimes it is just a consequence of the history of wavelets, because one of the main tools (namely, multiresolution analysis) only deals with dyadic wavelets.
In Section 12.1 we state results concerning the structure of a wavelet frame and its canonical dual frame. In particular, we show that the canonical dual of a wavelet frame might not have the wavelet structure. In case the given frame is not a Riesz basis, we know from Lemma 5.6.1 that other duals exist, and this naturally leads to the question whether a dual having the wavelet structure exists. We give a definition which expresses what we want, and return to concrete constructions in Chapters 13-14.
In Section 12.3 we review some recent results concerning frames generated by a function whose Fourier transform is a characteristic function.
Note that in this chapter we also will consider wavelet frames generated by more than one function. That is, we consider a finite number of functions
274
12. Dyadic Wavelet Frames
*01, • • • i 'Фп E L2(IR) and ask for
u {DjTk^2}j,kez U • • • U {DjTkipn}j^z (12.1)
to be a frame for L2(IR). A frame of this type is called a multiwavelet frame, and will usually be denoted by {PJ'Tfc^}j,fcGZ,£=i,...,n or
12.1 Wavelet frames and their duals
The canonical dual frame associated to a wavelet frame {'ipj,k}j,kez with frame operator S is given by {S~lipjik}j,ke^- Due to the difficulty in inverting S explicitly it is often hard to find the dual frame, but in the following example from [105] it can be done by direct computation. The example is important because it gives us an idea about which properties we can expect from the dual of a wavelet frame.
Example 12.1.1 Let {^j,k}j,kez be a wavelet orthonormal basis for L2(IR). Given e G]0,1[, we define a function 0 by
0 = ip + eDip.
We want to prove that {0j,k} j,k£Z is a Riesz basis and find the dual Riesz' basis. The idea is to consider 0 as a small perturbation of ip and use a stability result for frames to conclude that {03,k}j,kez is a Riesz basis. First, the commutator relation (2.11) shows that
= -eD^TkD^> = -eD’+1T2k4>-,	(12.2)
thus, given any finite scalar sequence {c^k}.
2
Cj,k ФР],к	@j,k)	—
j,k
Y,Cj,kDi+'T2k<p = e2 £ |cJ)fc|2; j,k	j,k
the last equality follows from {Di+lT2kip}j,ke% being a subfamily of the orthonormal basis {tpj,k}j,kez- Via the general perturbation result stated in Theorem 15.1.1 we see that {0j,k}j,kez is a Riesz basis for L2(IR). By the definition of a Riesz basis we can define a bounded invertible operator
U : L2(R) -> L2(R), u^k-.= e]tk.
Via Exercise 5.16, the frame operator for {0jtk}jtkez is S = UU*, so the canonical dual associated to {0j,k}3,ke% is
{8-1д,,к},,к& = {(ci*)-1 u~lejtk}^z = {(tr1)*(12.3)
We want to obtain a more concrete expression for the dual. In terms of the operator U, (12.2) means that
(1 - U)^,k = -е&+1Т2кф = -e^+i,2fc;
12.1 Wavelet frames and their duals
275
expanding an arbitrary f 6 L2(R) in the orthonormal basis {^j,k}j,kei it follows that
(7-t7)/ = -e j,kez
Thus, for f,gE L2(R),
(f,(I-U)*g) = {(I — U)f,g)
= -£ 52 j,/cGZ
= (/,-«52 Mj+i,2k)‘ipj,k)-
3,k£Z
It follows that
(I - U*)g = (I- Uyg = -e £	(12-4)
j ,k£Z
In particular, ||Z — U*\\ — e < 1, which implies that (t/*)-1 can be expanded in a Neumann series,
oo
n=0
now, (12.3) implies that the dual Riesz basis of {@j,k}j,ke% is
{OO	'l
.	(12.5)
n=o	) j,kez
We can go one step further. In fact, the action of I — U* on the functions ibj'kii'k G Z can be found via (12.4) using that {V^fch.fcGZ is an orthonormal basis; the outcome depends on к being even or odd:
(f - Ur^jt2k =	while (I - U*)4>jf2k+1 = 0, Vj, к 6 Z. (12.6)
In particular, via (12.5),
S^O^k+i = ^j,2k+i for all j,k e Z.
Also, for any к 0, the equations in (12.6) show that there exists a value of n G N for which
(I-U*)n^,2k=V,
so S^Oj^k is a finite linear combination of functions {^j,k}j,ke%,
S~l0jt2k = 4>j,2k + (I -U*)^,2k + ••• + (/- U*y^,2k = 4>j,2k - e4>j-i,k + • •  + 0.
276
12. Dyadic Wavelet Frames
For к = 0, we have
oo
S-1V’j,o =	(12.7)
п=0
In particular, the canonical dual frame of {@j,k}j,kez does not have the wavelet structure; the functions {S~10jik}j,kez do not even have the same norm. This is in contrast to the situation for Gabor frames and frames of translates, where we saw that the canonical dual has the same structure as the frame itself.
We also note that the above calculations show that there are other properties which are not inherited by the dual. For example, if we assume that the function has compact support, then also 6 has compact support, and all the functions {Oj,k}j,keZ have compact support. If we look at the dual then we obtain functions with compact support when к 0 0 because the elements in the dual frame are finite linear combinations of the functions in {tpj,k}j,kez in this case. However, for к = 0 the expression (12.7) shows that S~16jtQ is not compactly supported.	□
Example 12.1.1 is somewhat disappointing. Wavelet frames were introduced because their structure makes them convenient to work with, but as soon as we want to apply the frame decomposition we need to know the canonical dual. If the canonical dual has the wavelet structure, we can find it by calculating one single function (the generator) and then simply applying the operators DJTk,j, к G Z to get the other elements. But if the canonical dual does not have the wavelet structure, we have to find it as {S~1ipjik}j,ke^ =	that is, we have to apply S-1 to a
double-infinite set of functions, which is a much harder task. The heart of the problem is that, in general, the frame operator S does not commute with the operators DJTk, j,k € %, this implies that in general (see exercise 12.2 for a related statement)
{S~l^j,k}j,kez / {DjTkS~1ip}j^EZ‘
if ^kez is a Riesz basis, then {S l'ifj,k}j,ke^ can not be replaced by another sequence in the frame decomposition
j,kez
because the dual is unique, see Theorem 3.6.3. But if {ipj,k}j,ke7. is an overcomplete frame for which the dual is not a wavelet frame, it is natural to exploit the freedom offered by the redundancy and search for another dual which has the wavelet structure. We now give a formal definition expressing what we are interested in. For reasons that will become clear later (see e.g., Theorem 14.5.1) we state it for multiwavelet frames.
12.2 Tight wavelet frames 277
Definition 12.1.2 Consider two sequences of functions
C L2(IR) and ?/ii,... ,^n e L2(R).
iVe say that {DjTk^}jikez,£=i,...,n and {DjTk^}jikez,£=i,...yn are a pair of dual wavelet frames if both are Bessel sequences and
N
/ = EE {f, D^Tk^DjTk4>e, V/ € L2(K).	(12.8)
£=i j.fcez
That Bessel sequences {DJTkip£}jikEz,£=i,...,n and {DjTk^E}j,kez,£=i,..,n are frames if they satisfy (12.8) follows from Lemma 5.6.2. A pair of dual wavelet frames is called sibling frames in [90] and bi-frames in [112].
Daubechies and Han gave in [110] an example of a wavelet system {Vj.aJj./cgz for which the canonical dual does not have the wavelet structure; however, there exist infinitely many functions гр for which {^j,k}j,kez. is a dual frame. The generator of this specific frame has the property that гр = xk for a compact subset К of R; frames of this type are the subject of Section 12.3.
A characterization of all pairs of dual wavelet frame pairs was obtained by Frazier et al. [140]:
Theorem 12.1.3 Let гр!,... ,грп,гр1,. • • .'Фп 6 L2(IR) and assume that {DjTkipi}jikez,i=i,...,n and {DjT^i}jtkez/=it...tn are Bessel sequences. Then {DjTk^^}jikeZ,£=i,...,n and {DjT^e}jikEZ,£=i,...,n are a pair of dual wavelet frames if and only if the two equations
n	--------
EE ^(2jt)^(2^7) = 1 e=i jez
and n oo	—-------------
52 52^(2j7)^(2j'(7 + <?)) = 0 for all odd integers q 1=1 j=o
hold a.e.
Chui and Shi have extended Theorem 12.1.3 to wavelet frames with arbitrary dilations and translations, see [96].
12.2 Tight wavelet frames
As we have discussed in Section 5.7, tight frames are very convenient to work with because the frame decomposition can be applied without any cumbersome inversion of the frame operator. The dual of a tight frame
278
12. Dyadic Wavelet Frames
{^j,k}j,kez with frame bound A is simply {-дФз,к}д,ке%', that is, in contrast to the situation for general wavelet frames, the canonical dual of a tight wavelet frame automatically has wavelet structure. The frame decomposition states that
f = 2	v/gl2(R).
Similar to the Gabor case, the functions ip generating a tight wavelet frame can be characterized based on the characterization of dual wavelet pairs. We leave the details to the reader (Exercise 12.3) and also note that a direct proof can be found in e.g., [173].
Theorem 12.2.1 A function ip 6 L2(JR) generates a tight wavelet frame {^j,k}j,kEZ with frame bound A if and only if the equations
£МЪ)|2 = A,	(12.9)
'0(2j7)'0 (2-7(7 4- q)) = 0 for all odd integers q (12.10) з—о
hold a.e..
Note that the only difference between the conditions for {^j,k}j,kez being a tight frame with frame bound equal to 1, and the characterization of wavelets on page 76, is that the condition \\ip\\ = 1 does not appear in Theorem 12.2.1.
Chapter 14 will deal with general constructions of tight wavelet frames via an extension of the classical multiresolution analysis scheme.
12.3 Wavelet frame sets
Theorem 11.2.3 gives a sufficient condition for a function ф g L2(R) to generate a wavelet frame	expressed in terms of the Fourier
transform 7^. For special classes of functions ip we can give simpler conditions for ip generating a wavelet frame; one natural choice is to consider functions ip for which ip is an indicator function for a Lebesgue measurable set К in JR. In order for у к to belong to L2(JR) we assume that К has finite Lebesgue measure:
Definition 12.3.1 A Lebesgue measurable set К in JR is called a frame wavelet set if |7<| < 00 and the function ip defined by ip = xk generates a wavelet frame {^j,k}j,kez for L2(JR).
We will give a short description of results obtained by Han [162], respectively Dai et al. [104]. We begin with some definitions:
12.3 Wavelet frame sets
279
Definition 12.3.2 Let К be a measurable set in R with finite measure. pfe say that
(i) x,y 6 R are 5-equivalent if there is an j EZ such that
x = 2^y.
For x G К, the number of elements у € К which belong to its 6-equivalence class is denoted by &k(x). Finally, let
K(5, к) := {x G К : 6K(x) = k}, к EM
(ii) x,y G R are т-equivalent if there is an к EZ such that
x = у 4- к.
For x G K, the number of elements у E К which belong to its r-equivalence class is denoted by tk(x). Finally, let
K(t, k) := {xtK: tk(x) = k}, fc EM
Using the above notation, Dai et al. were almost able to characterize frame wavelet sets:
Theorem 12.3.3 Let К be a Lebesgue measurable set in R with finite measure. Then the following holds:
(i) К is a frame wavelet set if Uyez2-?K(r, 1) = R (up to a null-set) and there exists M G N such that	and K(r,m) are null-sets for
m > M; in this case one is a lower frame bound for {ipj k}j kez and M5/2 is an upper frame bound.
'(ii) If К is a frame wavelet set, then Ujez2J/C = R (up to a null-set) and there exists M G N such that	and K(r,m) are null-sets for
m > M.
For frame wavelet sets generating a tight frame, a complete characterization is obtained:
Theorem 12.3.4 A Lebesgue measurable set К in R with finite measure is a frame wavelet set generating a tight frame if and only if the following conditions hold:
(i) Ujgz2-71C = R (up to a null-set);
(ii) for some m > 1 we have К = К(т, 1) = К(5, m).
In case (i) and (ii) are satisfied, the frame bound for	equal
to m.
280
12. Dyadic Wavelet Frames
Let us show how the conditions in Theorem 12.3.4 can be reformulated. The condition К = К (г, 1) means exactly that for 7 E R, the point 7 4- k belongs to К for at most one value of к G Z; or, expressed differently, that
+ k) <1, a.e. 7 £ R.	(12.11)
k£Z
Now assume that
2J К = IR, and for some m G N, К = К(6, тп). (12.12) jez
Then, given 7 G IR there exists j' G Z such that 2-J 7 G K. The J-equivalence class of 2_J 7 contains exactly m elements, so
£2 X2>k(?) = a.e. 7 £ R.	(12.13)
jez
Similarly, one proves that if (12.13) holds for some m G N, then (12.12) holds. Thus we have obtained an equivalent formulation of Theorem 12.3.4:
Theorem 12.3.5 A Lebesgue measurable set К in JR with finite measure is a frame wavelet set generating a tight frame if and only if (12.11) and (12.13) are satisfied for some m > 1.
In the case where К is a finite union of closed intervals, the sufficiency of the conditions (12.11) and (12.13) was obtained by Han a couple of years before Dai et al. proved Theorem 12.3.4. We illustrate the use of Theorem 12.3.5 with some examples:
Example 12.3.6 (i) Let К = [— |, — |[U]|, |]. Then, except for 7 = 0 there exists exactly one value of j G Z such that 7 G WK, so (12.13) is satisfied with m = 1. Equation (12.11) is also satisfied, so К is a frame wavelet set, which generates a tight frame with frame bound one.
(ii) Similarly, for n = 1,2,..., the set К = [-1[U]	, |] is a frame;
wavelet set, which generates a tight frame with frame bound n.
(iii) Let К =	|[. Then
ши = 4.444
and	= Iup to a null-set. Also, for m > 2 we have
A'(5, m) = K(r, m) = 0.
Thus, by Theorem 12.3.3, К is a frame wavelet set.	□
12.4 Frames and multiresolution analysis 281
12.4 Frames and multiresolution analysis
The classical definition of multiresolution analysis was introduced with the purpose to construct orthonormal bases for L2(K). There are several natural ways to extend the scope to construction of Riesz bases or frames. The biorthogonal multiresolution by Cohen, Daubechies and Feauveau [99] delivers Riesz bases {^j,k}j,kez,	via the construction of two mul-
tiresolution analyses. Another option is to consider Definition 3.8.2 without modifications and ask for the existence of a function ip G Vo for which {V’j.fch'.fcGZ is a Riesz basis or a frame for L2(JR). This idea was elaborated by Zalik. In [289] he characterizes all functions ip which appear via some multiresolution analysis and generate Riesz bases; in [290] he considers the similar question for frames, and obtains equivalent conditions for a frame-generating function ip to be associated with a given multiresolution analysis.
Other authors have constructed dyadic wavelet frames by modifying the original definition of a multiresolution analysis. In the next chapter we discuss frame multiresolution analysis as defined by Benedetto and Li, and in Chapter 14 we give a treatise of another approach by Ron and Shen. Common for them is that they keep the conditions in Definition 3.8.2, except condition (v).
12.5 Exercises
12.1	Assume that К is a frame wavelet set, and let 3 6 L2(JR) be a function with support on К. Assume that there exist constants C, D > 0 such that C < |0| < D. Prove that the function ip 6 L2(K) defined by ip = 3%k generates a wavelet frame.
12.2	Let {DJTkip}jtkez. be a frame with frame operator S. Prove that S commutes with D\j 6 Z, and thereby that
{S-lD^}jtkeZ = {DiS-'T^}^.
This result shows that it is enough to calculate a single-infinite family of elements in order to find the dual frame (compare to the discussion on page 276).
12.3	Prove Theorem 12.2.1 via Theorem 12.1.3.
13
Frame Multiresolution Analysis
The introduction of multiresolution analysis by Mallat and Meyer was the beginning of a new era; the short descriptions in Section 3.8 and Section 4.2 only give a glimpse of the research activity based on this new tool, aiming at construction of orthonormal bases
As described in Section 4.2 and Section 12.4, the “classical” theory has been extended in different ways with the purpose of removing some of its constraints; as an example, we mention the biorthogonal multiresolution analysis, which leads to constructions of Riesz bases and their duals. In this chapter and the next we go one step further and extend the multiresolution scheme in such a way that we can construct over complete wavelet frames. It is important to notice that we insist on the key idea of multiscales; they make the constructions attractive from the computational aspect, as a reader familiar with multiresolution analysis will know.
This chapter will deal with frame multiresolution analysis as defined by Benedetto and Li; here the condition (v) in Definition 3.8.2 is simply replaced with the condition that {Ткф}кея is a frame for Vq. This seemingly innocent change causes many technical difficulties, but under certain conditions we are actually able to construct wavelet frames from here.
Frame multiresolution analysis is not the most general way to obtain frames via multiscale techniques, but it provides us with a natural link from the classical constructions described in Section 3.8 to the more advanced theory presented in Chapter 14.
This Chapter is independent of Chapters 11-12. However, the results for frames of translates (especially Theorem 7.2.3) will play an important role.
284	13. Frame Multiresolution Analysis
13.1 Frame multiresolution analysis
Frame multiresolution analysis was introduced by Benedetto and Li [23], [22]. The purpose of the theory is to construct wavelet frames for L2(I&) of the form {2J/2'0(2-7\r — k)}jtkE%. We will use the notation introduced at the beginning of Chapter 12 and denote such a frame by {D^Tk^} jikEz or
We introduce some terminology before we define frame multiresolution analysis. The interval ] — j, j[ is identified with the torus T, and the class of 1-periodic functions on К whose restriction to ] — j, |[ belongs to L2(—|) is denoted by L2(T). Similarly, L°°(T) consists of the bounded measurable 1-periodic functions on R. With this notation, L°°(T) C L2(T). We note that L2(T) and L°°(T) actually consist of equivalence classes of functions which are identical almost everywhere, so when we speak about pointwise relationships between functions it is understood that they can only be expected to hold almost everywhere. In the entire section we will not mention this explicitly, i.e., we will not follow the equations by “a.e.”.
One of the main tools will be Fourier expansions of functions in L2(T). Using the complex exponentials Ek(x) = e^ikx, the pourier series of a function f e L2(T) will be written
i
/ = ^cfcEfe, where ck = f(x)E_k(x)dx.
kez
Remember also the commutator relations from Section 2.5:
TkDj = D*Tvk, &Тк=Т2-,к&,
7Ta = E_a:F, EEa = TaE, ED~D~}E.
Formally, a frame multiresolution analysis is defined as a multiresohb tion analysis, with the condition “{Ткф}кЕ% is a orthonormal basis for Vo” replaced by a frame condition:
Definition 13.1.1 A frame multiresolution analysis for L2(R) consists of a sequence of closed subspaces {Vjjjez of L2(R) and a function ф G Vo such that
(i)	• • • V-i C Vo C VS • • •.
(ii)	= L2(R) and AjU, = {0}.
(iii)	V3 = DWq.
(iv)	f ev0^Tkfe Vo, Vk e z.
(v)	{Ткф}ке% is a frame for Vo.
In principle one can start the construction of a frame multiresolution analysis with a subspace Vo C L2(R) satisfying (i)-(iv), and then search
13.1 Frame multiresolution analysis 285
for a function ф such that {Ткф}ке2 is a frame for Vo. In practice however, the starting point is always a function ф for which {Ткф}ке% is a frame sequence, and the spaces Vj are defined by
Vj :=	= sp^{BJT^}fcGz, j e Z. (13.1)
We first note that in this case it follows that TijVj = {0} without any extra assumption; this is actually a classical result from multiresolution analysis! In [106] for example, it is proved that F\jVj = {0} under the assumption that {Ткф}ке% is a Riesz sequence, but the first step in the proof is to observe that then {Ткф}ке% is a frame sequence, and this is all that is needed for the argument. A more general result obtained by deBoor, DeVore and Ron [31] shows that even the frame condition can be removed:
Lemma 13.1.2 Let ф 6 L(i) 2(K) and define the spaces Vj by (13.1). Then = {0}.
With this in mind it is convenient to formulate a shorter definition of a frame multiresolution analysis, where the redundancy in Definition 13.1.1 is removed (see also Exercise 13.1):
Definition 13.1.3 A function ф 6 L2(K) generates a frame multiresolution analysis if {Ткф}ке% is a frame sequence and the spaces {Vjjj^z defined by (13.1) satisfy the conditions
(i) • • • V-i C Vo C Vi • • •.
(ii) U~Vj = L2(K).
We will consequently refer to this version of the definition.
Two major questions concerning frame multiresolution analysis are;
(i) Under which conditions does a function ф 6 L2(K) generate a frame multiresolution analysis?
(ii) If ф 6 L2(K) generates a frame multiresolution analysis, can we construct a function such that {2J/2,0(2Jx — k)}ke% is a frame for L2(K)?
It turns out that sufficient conditions for ф to generate a frame multiresolution analysis can be found by small modifications of results concerning “classical” multiresolution analysis, so we will only give a relatively short description of this part in Section 13.2. The question (ii) is more complicated, and we will treat it in detail in Section 13.4.
286	13. Frame Multiresolution Analysis
13.2 Sufficient conditions
The purpose of this section is to find sufficient conditions for a function ф E L2(JR) to generate a frame multiresolution analysis. We are mainly interested in the case where {Ткф}ке% is an overcomplete frame sequence; note that Proposition 7.3.6 puts some restrictions on ф in order for this to happen.
As starting point we will consider a function ф for which {Ткф}ке% is a frame sequence; recall that Theorem 7.2.3 gives an equivalent condition for this in terms of the function
ф(?) := + £)|2-
к GZ
In order for the spaces Vj defined by (13.1) to satisfy (i) and (ii) in Definition 13.1.3 it is natural to follow the approach used in the classical multiresolution analysis. In [106] the density of UV) in L2(K) is obtained by assuming that
(i)	{Ткф}ке2 is a Riesz sequence, and
(ii)	ф is bounded and continuous in 0 with ф(0)	0.
The first step in the proof in [106] is to notice that a Riesz sequence is a frame sequence; no special properties for Riesz sequences are needed. That is, we can replace the word “Riesz sequence” in (i) by “frame sequence”. Concerning (ii), we observe that by Theorem 7.2.3, the function ф is automatically bounded when {Ткф}ке% is a frame sequence. Also, the condition of continuity of ф in 0 with a non-vanishing function value can be replaced by
(iii)	\ф\ > 0 on a neighborhood of zero.
Now we only need a condition ensuring that Vj C Vj+i for all j G Z. For this purpose we first state some properties for V3:
Lemma 13.2.1 Let ф 6 L2(JR) and assume that {Ткф}ке2 is a frame sequence with frame bounds A,B. With Vj,j 6 Z, defined as in (13.1) the following holds:
(i)	{В^Ткф}ке1 is a frame for Vj with frame bounds A,B.
(ii)	A function f G L2(R) belongs to V3 if and only if f = ^2ке%скЕ^ТкФ for some {ck}kez € ^2(Z).
(iii)	A function f G L2(K) belongs to Vj if and only if there exists a 1-periodic function F G L2(T) such that
(13.2)
13.2 Sufficient conditions
287
If f e Vj, the function F is uniquely determined on all у for which Ф(7) 0 0; if Ф(7) = 0 one can choose F(y) = 0.
proof. Since D is unitary, (i) follows from Lemma 5.3.3. (ii) is a consequence of (i) combined with Theorem 5.5.1. For the proof of (iii), let f G Vj] taking the Fourier transform of the expression in (ii) and using the commutator relations, we have (see Lemma 7.2.1)
f = т^скТкф = D-i ^скЕ-кф-fcez	fcez
This implies that
/(2>7) = 2-^2(D>/)(7) =
fcez
Thus we have the formula (13.2) with F(y) = 2_J/2 скЕ-к(у). On the other hand, if f G L2(JR) and F € L2(T) satisfy (13.2) for some j G Z, let us denote its Fourier coefficients for F by {dk}ke% and define ck = 2^2dk] then f = ^кегс-кЕ3Ткф G Vj.
For the last part of (iii) we note that if Ф(т) 0 0 for some 7, then there exists к G Z such that ^(7 4- к) 0; since
f^^ + k)) = F^ + k)^ + kfi
and F is assumed to be 1-periodic, it follows that
ФЬ + к)
If Ф(7) = 0 for some 7, then ^(7 4- k) — 0 for all к G Z. The equation (13.2) is satisfied no matter how ^(7) is defined, but if we want F to be 1-periodic we must require F(y 4- k) = F(y),k G Z. One choice is to take F(7) = 0 for all 7 for which Ф(7) — 0.	□
Conditions for Vj C V)+i are given in the following lemma:
Lemma 13.2.2 Assume that ф G L2(K) and that {Ткф}ке% is cl frame sequence. Define the spaces Vj by (13.1). Then the following conditions are equivalent:
(i) Vj C Vj+i for all j G Z.
(H) Vo C Vi.
(iii) There exists a 1-periodic function Ho G L°°(T) such that
^(7) = Яо(7/2Й7/2).	(13.3)
If (13.3) is satisfied, the functions Hq and Ф are related by
Ф(7) = |Яо(7/2)|2Ф(7/2) 4- |Я0(7/2 4- 1/2)|2Ф(7/2 4- 1/2).	(13.4)
288	13. Frame Multiresolution Analysis
Proof. (iii)=>(i). Assume that (iii) is satisfied, and let f G Vj. Using Lemma 13.2.1 (iii) and the assumption, there exists a function F G L2(T) for which
/(2>+17) = F(27)0(27) = F(27)ffo(7)^(7)-
The function 7 i-> F(27)H0(7) is 1-periodic and belongs to L2(T) because F e L2(T) and Ho G L°°(T), so Lemma 13.2.1 (iii) shows that f G Vj+1.
For (i)=>(ii) there is nothing to prove. To prove that (ii)=>(iii), assume that (ii) is satisfied. Then ф G Vo C Vj, and В~}ф G Vo. By Lemma 13.2.1 (iii) there exists a 1-periodic function Hq G L2(T) such that 77P-1(/)(7) = Я0(?)Ф(7); since ВВ~1ф('у) = 21/2ф(27), a slight redefining shows the existence of a 1-periodic function Ho G L2 (ТГ) such that
^(27) = Н0(7И7).	(13.5)
Let us choose Ho such that Ho(7) = 0 if $(7) = 0. We now prove (13.4) and that Ho is bounded. First, (13.5) implies that
Ф(7)
E |<£(7 + fc)|2
E
If we split the sum into sums over even integers 2k, к G Z and odd integers 2k + 1, к G Z, and use the periodicity of Ho, we arrive at
$(7)
E
7 4- 2fc 4- 1 • 7 4- 2fc 4-1
—о—m—б—

l^o(7/2)|2E|0(7/2 + fc)|2 fcez
+|Я0(7/2 + 1 /2)|2 E |0(7/2 4- 1/2 4- fc)|2
|Я0(7/2)|2 Ф(7/2) 4- |Я0(7/2 + 1/2)|2Ф(7/2 4-1/2).
This proves (13.4) To show that Ho is bounded we consider 7 G R such that Ф(у) £ 0. Let A, В denote frame bounds for {Ткф}к€£- By Theorem 7.2.3 we have A < Ф(7) < В, and via (13.4) this gives that
в > ф(27) > |Я0(7)|2 Ф(7) > А |Но(7)|2.
Thus Но G L°°(T) as desired.
An equation of the type (13.3) is called a refinement equation; we say that ф is refinable. In case of a classical multiresolution analysis, where {Ткф}ье%
13.2 Sufficient conditions 289
is an orthonormal system or a Riesz sequence, the function Ho satisfying (13.3) is unique; this follows from Lemma 13.2.1 (iii) because Ф is bounded away from zero by Theorem 7.2.3. For a general frame multiresolution analysis, several choices for Hq might be possible because $(7) might be zero for 7 belonging to a set with positive Lebesgue measure. A convenient choice is to define Яо(?) = 0 if $(7) = 0, as we already did in the proof of Lemma 13.2.2; the function Hq obtained this way is called the two-scale symbol for the frame multiresolution analysis.
By collecting all the obtained information we obtain a sufficient condition for ф to generate a frame multiresolution analysis:
Theorem 13.2.3 Suppose that ф G L2(K), that {Ткф}ке% ls a frame sequence, and that \ф\ > 0 on a neighborhood of zero. If there exists a function Hq G L°°(T) such that
0(7) = Ho(	(13.6)
then ф generates a frame multiresolution analysis.
The statement of Theorem 13.2.3 corresponds exactly to the formulation of the similar result for multiresolution analysis in [288], Theorem 2.13.
Example 13.2.4 Define as in Example 7.2.4 the function ф via its Fourier transform,
0(7) = X[-a,a[, for some a e]0, |[.
We have already seen that {Тьф}ке2 is a frame sequence. Note that
^(2?) = X[-f ,f[(7) = X[-f ,^[(7)^(7)-
For |7| < 1, let
#0(7) =
extending Hq to a 1-periodic function we see that (13.6) is satisfied. By Theorem 13.2.3 we conclude that ф generates a frame multiresolution analysis.
Given a continuous non-vanishing function 9 on [—a, a], we can generalize the example by considering
^(7) := 0(7)X[-<x,a[(7)-
Defining
{0(271 if - p « Г «(7) У I 2 > 2 l> 0 if7 6[-|,-f[U[f,i[>
and extending Hq periodically, it again follows that ф generates a frame multiresolution analysis.	□
290	13. Frame Multiresolution Analysis
13.3 Relaxing the conditions
In Theorem 13.2.3 all the conditions in frame multiresolution analysis were derived on the basis of a function ф generating a frame sequence {Т^ф}^^2-In Chapter 14 we will consider another multiresolution scheme, proposed by Ron and Shen [246], where it is not assumed that {Тьф}ке% is a frame sequence. For this reason we now show that (i) and (ii) in Definition 13.1.3 can be satisfied without any frame assumption.
The basic idea of Ron and Shen is to replace the frame condition on with the condition that ф satisfies a refinement equation. Recall from Lemma 13.2.2 that if the spaces Vj in (13.1) are nested and is a frame sequence, then ф satisfies a refinement equation. We now prove that a refinement equation is enough to imply that Vj are nested, and Example 13.3.4 will show that nothing guarantees that {Ткф}ке% is a frame sequence; thus, the idea of Ron and Shen is in fact more general.
Lemma 13.3.1 Assume that ф E L2(JR) and that {Тьф}ке% a Bessel sequence. Define the spaces Vj by (13.1). Then the following holds:
(i) If ip e L2(K) and there exists a function F G L°°(T) such that = ^(7)^(7), then ip 6 Vj.
(ii) If there exists a function Hq G L°°(T) such that
ф(2у) = Н0(у)ф(у),	(13.7)
then Vo C Vi.
Proof. Let ip G L2(K) and assume that for some F G L°°(T) we have = ^(7)^(7). Writing the Fourier series of F as F =	we
have (see Lemma 7.2.1)
= F(j) = ^2 скЕкф = СкТ-кФ‘
Since Dip = FD-1ip, this shows that D~xip = y/2 скТ-ьФ € Vo, i.e., ip G DVq = Vj. This proves (i). (ii) follows from here because Vj is closed and invariant under integer-translations.	□
The condition (ii) in Definition 13.1.3 can also be derived without assuming ф to be a frame sequence. This follows from a result by deBoor, DeVore and Ron [31]:
Lemma 13.3.2 Let ф G L2(K) and assume that the spaces Vj in (13.1) are nested. If \ф\ > 0 on a neighborhood ofO, then UjVj is dense in L2(IR).
Via Lemma 13.3.1 and Lemma 13.3.2 we have:
13.3 Relaxing the conditions
291
'Theorem 13.3.3 Let ф £ L2(IR). Assume that (13.7) is satisfied for a function Hq 6 L°°(T) and that |^| > 0 on a neighborhood of 0. Then the spaces Vj defined in (13.1) satisfy the conditions (i)-(iv) in Definition 13.1-1.
In this chapter we will always assume that {Ткф}кё£ is a frame sequence, in which case Theorem 13.3.3 equals Theorem 13.2.3. The role of Theorem 13.3.3 will be clear in Chapter 14; for now, we just give an example, which shows that it can actually happen that all the conditions in Definition 13.1.1 except (v) are satisfied.
Example 13.3.4 Assume that ф G L2(R) generates a classical multiresolution analysis for L2(]R), where {Ткф}ке% is an orthonormal basis for Vq. Let
ф = ф + Т}ф.
Then
{Tk<t>}ke% — {Tk+i<f> 4- Ткф}ке%-
From Example 5.4.6 we know that Vq = spanl!^}/^, and that {Ткф}ке% is not a frame for Vq. However, all the other conditions for ф generating a frame multiresolution analysis are satisfied.
The function ф satisfies a refinement equation. In fact, since ф generates a multiresolution analysis, it is known from the classical theory or from Lemma 13.2.2 that there exists a function Hq E L°°(T) such that
0(27) = #o(7)0(7);
using that
Ф = (1 + Е-^ф,
it follows that
0(27) - (1 + е-4^)Я0 (7)0(7)
I _1_ p-47ri7	z
= Т77^7Яо(7)0(7) = : Яо(7)0(7), 7 И 1 + Z.
□
The reader can verify that Hq g L°°(T).
292	13. Frame Multiresolution Analysis
13.4 Construction of frames
The next question is whether a frame multiresolution analysis can be used to construct a frame for L2(K). We prove in Theorem 13.4.5 that an extra condition is needed in order to assure this.
Assume that ф 6 L2(K) generates a frame multiresolution analysis, and let Wj denote the orthogonal complement of Vj in Vj+1. Exactly as in the case of a multiresolution analysis this gives the orthogonal decomposition
L2(R)=®^.	(13.8)
jGZ
All we need in order to construct a frame for L2(K) is a function ф 6 L2(K) for which {Ткф}ке2 is a frame for Wo. This follows from the observation that the spaces Wj are related by the same dilation property as we have for Vj:
Lemma 13.4.1 Assume that ф 6 L2(K) generates a frame multiresolution analysis. Then the following holds:
(i) Wj = D3Wq, Vj e Z.
(ii) 1ффЕ Wq generates a frame {Ткф}ке2 for Wo, then for all j G Z, the family {В3Ткф}ке1 is a frame for Wj, and {D3Tk^}jtkez ™ a frame for L2(K); these frames have the same frame bounds as {Ткф}ке%-
Proof. For the proof of (i), let f € Wo. Then f 6 Vj, so DJ f 6 Vj+i-Furthermore, /±Vo, so since DJ is unitary, DJ f_LDJVo = Vj. This proves that DjWq C Wj; the proof of Wj C DJW0 is similar.
Now assume that {Ткф}ке% is a frame for Wo with frame bounds A, B. Then Lemma 5.3.3 shows that {В3Ткф}кЕ.1 is a frame for
spanlP-’Tfc^lfcez = DjW0 = Wj,
also with frame bounds A, B. Let f € L2(K). Denoting the orthogonal projection of L2(K) onto Wj by Qj, we have by (13.8) that f — ^jezQif and
н/п2 = Е1шн2-
The last part of the proof follows from this combined with the observation that
a ||Q,7II2 <Ек<з,/,лЗД|2 = Е|(/,1)>ад)|2 <в iiQj/ц2- □ kez	fcez
In the classical case where ф generates a multiresolution analysis, we know that there always exists a function ф 6 Wo such that {Ткф}ке% Is an orthonormal basis for Wo and {V’j,k}J,kez is an orthonormal basis for
13.4 Construction of frames 293
L2(K). The corresponding result for a frame multiresolution analysis is more complicated: there might not exist a function ф 6 L2(K) for which {TfcV’jfcGZ is a frame for Ил0. Equivalent conditions for the existence of such a function were found by Benedetto and Treiber [25]; we need some preparation before we present their result in Theorem 13.4.5.
The first step is to characterize the space Wq. In Lemma 13.2.1 we have seen that if {Ткф}ке% is a frame sequence and F E L2(T), then the function f defined by /(27) = ^(7)^(7) belongs to Vi. An extra condition implies that f G Wo:
Lemma 13.4.2 Assume that ф E L2(K) generates a frame multiresolution analysis with two-scale symbol Hq G L°°(T). Let F 6 L2(T) and define f G Vi by = F(7)0(7). Then the following holds:
(i) {/,Ткф) = 2 [F&TT0 + T1/2 (ЯФ2Ж)] E2k.
(ii) f G Wq if and only if
HoF$ + Т1/2(ЛоРФ) = 0 on [0,1[.	(13.9)
Proof. For к G Z we use the Fourier transform and (13.3) to obtain that (/,T^) = (f,E_k&
= (F(-/2)^(-/2))E_fc(-)^o(-/2)^(-/2))
= f Е(у/2)ф(у/2)е-^-гН0(у/2)ф(-у/2)с1У
J —00
fOO
= 2 /	F(7)|^7)|2F^)e4^d7.
The function 7 i-> Г(7)Я0(7)е47ггА:7 is 1-periodic, so (we ask the reader to justify the rearrangements)
(/, Ткф) = 2 Л (f(7 + n)|<£(7 + п)|2Я0(7 + п)е47Г!^+п’) d7
= 2 P F^^H^e^dy.
Splitting the integral in two, we can continue with
294
13. Frame Multiresolution Analysis
= 2 Jo
+2 [2 F(-y - 1/2)|Ф(7 - 1/2)|2Я0(7 - l/2)e4’rifc(7-1'/2^7 Jo
= 2^ [р(7)|Ф(7)|2Жт) + Г1/2 (г(7)|Ф(7)12Яо(7))] ^(7)^7-
This proves (i). By definition, / e Vi, so in order to prove (ii) we have to prove that / is orthogonal to Vo = span{Tfc^}fcGz if and only if (13.9) is satisfied; but this follows from {V^B2k}kei being an orthonormal basis for b2(0,l).	□
Note that the function ЯоЯФ4-Т1/2(ЯоРФ) is ^-periodic. Thus, if (13.9) holds on [0, l/2[, then it holds on JR.
We can now give a condition for the existence of a frame {Tfc'0}fcGZ for PVq. As standing notation in the rest of the chapter, we let F 6 L2(T) and
G Vi be related by
^(2?) = ^(7)^(7)-
(13.10)
Also, let
ф(ч) = 5Ll^(7 + fc)|2-
Proposition 13.4.3 Assume that ф 6 L2(IR) generates a frame multiresolution analysis with two-scale symbol Ho 6 L°°(T). Let F 6 L°°(T) and define ф 6 Vi by ^(27) = ^(7)^(7). If there exist Gq,Gi 6 L°°(T) such that the three equations
7^F$ + T1/2(HoF$)	=	0,	(13.11)
HqGq^ + FG^	=	Ф,	(13.12)
Т1/2(ЯОФ)С?О4-Т1/2(ЯФ)(71	=	0,	(13.13)
are satisfied on T, then {Ткф}ке% is a frame for Wq.
Proof. The first step is to show that {Ткф}ке1 is a frame sequence. Let A, В denote frame bounds for {Ткф}к^. We note that an argument similar to the proof of (13.4) applies to Ф; it gives that
Ф(27) = |Г(7)|2Ф(7) + 1^(7 + 1/2) |2Ф(7 + 1/2).	(13.14)
We want to apply Theorem 7.2.3, so we have to show that outside its zero set, the function Ф is bounded away from zero and above. Since we have assumed that F is bounded, it immediately follows from (13.14) and
13.4 Construction of frames
295
Theorem 7.2.3 that Ф is bounded above. In order to prove that Ф is bounded below, it is enough to estimate Ф(2т) for 7 G [0, |[. We examine four cases separately, so let us define
T1	:=	{7 € [0,Ф(7)	= 0,	Т1/2Ф(7)	=	0}.
T2	:=	{7 € [0,1[:	Ф(7)	> 0,	Т1/2Ф(7)	>	0}.
Тз	:=	{7 e [0,1[:	Ф(7)	> 0,	Г1/2Ф(7)	=	0}.
T4	:=	{7 6 [0,1[:	Ф(7)	= 0,	Т1/2Ф(7)	>	0}.
If 7 € Ti, then Ф(27) = 0, so this case is okay. If 7 6 T2, it follows by (13.14) that
Ф(27) > A (|F(7)|2 + |Г(7 + 1/2)|2).	(13.15)
Furthermore, we see by (13.12) that the two equations
Ho(7)^o(7)+F(7)(71(7) = 1,	(13.16)
T1/2(HoGo)(7)+r1/2(F(71)(7) = 1,	(13.17)
hold. Now assume that for some c G]0,	an<^ some 7^2 we have
e2
|T1^WI" ЙЖ	(13-18)
Then |(T1/2F)(7)G?i (7)| < e2, and (13.13) implies that
|(Т1/2Яо)(7)Со(7)| <e2.	(13.19)
Therefore at least one of the following two options holds:
(i) |Т1/2Я0(7)| < e;
(ii) |G0(7)| < e.
We will use (13.15) to obtain a lower bound for Ф(27) in each of these cases separately. In case (i), (13.17) gives
ITWGOWI = |1 - (T1/2Ho(7))(T1/2Go(7))| > 1 - e||Go||oo, and therefore
|T1/2F(7)| >	(13.20)
1 + ||Cri lloo
The equations (13.18) and (13.20) give a contradiction if 1-ellGolloo e2 1 + HGiHoo 1 + IIGrlloo’
296
13. FYame Multiresolution Analysis
i.e., if
-IIGolloo + 4/l|Goll2oo + 4
2
Thus, in case (i) the inequality (13.18) shows that we have a lower bound on |Tly/2F|; by (13.15) this gives a lower bound on Ф(27). In the case (ii) we apply (13.16) to get
|F(7)G1(7)| = |1 - Яо(7)Со(7)| > 1 - |Я0(7)| |Go(7)l > 1 - «1|Я0||оо, which by the choice of e implies that
~ ell-^olloo _______1__________1_____
1	1 + HGJIoo 1 4- HHolloo 1 4- Halloo’
Thus |F(y)| is bounded below in case (ii), and again we conclude via (13.15) that Ф(2у) is bounded below.
If 7 6 T3, then
Ф(27) = |Я(7)|2Ф(7).
By (13.12) and (13.11) we have
Ho(7)Go(7)4-F(7)G1(7) = 1, W(7) = 0.	(13.21)
The last equation is satisfied if F(y) = 0 (leading to Ф(2у) = 0) or if Я0(7) = 0; in the latter case, the first equation in (13.21) gives F(7)Gi(7) = 1, and therefore |F(7)| >	. Thus
ф(27) - м?
as desired.
Now let 7 e T4. Then Ф(2у) = |F(y 4- 1/2)|2Ф(у 4- 1/2). Translating (13.12) we obtain that
Т1/2(Я0С0Ф)(7) + Г1/2(ЯС1Ф)(7) = Т1/2Ф(7).	(13.22)
By (13.11) we have Tly/2(FoF)(7) = 0. If TI/2F(7) = 0 we have Ф(2у) = 0; otherwise,	= 0, and (13.22) gives that T1/2(FG?i)(7) = 1. As in
the previous case this leads to
ф(27) -
This completes the analysis of the four separate cases: now we know that Ф is bounded away from zero outside the set where it is equal to zero, so {Tk^}kez is a frame sequence. The rest of the proof will show that span{Tfc'0}fcG2 = Wo- For this purpose we first rewrite the equations (13.12) and (13.13). If 0(7) 0 0, then Ф(у) £ 0, and we can multiply (13.12) with the obtained equation clearly also holds if 0(7) = 0, i.e., we have
Ho(7)G,o(7)<^(7) 4- F(7)G! (7)^(7) = 0(7), 7 € R (13.23)
13.4 Construction of frames 297
g/e can rewrite (13.13) in the same way to obtain
Т1/2(Яо^)(7)Со(7) + T1/2(F^)(7)G1(7) = 0, 7 € R;
applying the operator T_1/2 this can also be written
H0(7)^(7)G0(7 4-1/2) 4- F(7H7)G4(7 4-1/2) = 0, 7 G R. (13.24) Let {с/cjkez and {dkjkez denote the Fourier coefficients for Go and G^; then
<?o(7) = Ecfce2”^, Gi(7) = fcez	fcez
Note that
Go(7 + 1/2) = 52 cke2nik^+1^ = £ ck(-l)ke2*ik'1 fcez	/cgz
with a similar calculation valid for Gi. Inserting these expressions in (13.23) and (13.24) and recalling (13.10) and that Яо(7)0(7) = 0(27) by (13.3), we obtain the equations
0(7) = 0(27) У2^е2”^+0(27) £^e2"^ fcez	fcez
and
0 = 0(27)^Cfc(-l)'=e2’ri'!-' + 0(27)524(-l)fce2’ri':7. kez	kez
By addition (respectively subtraction) it follows that
0(7) = 20(27) ^2 c2fce2«2^ +20(27)	d2fce2”2^
/cgz	fcez
and
0(7) = 20(27) ^c2k+1e2^2k+^ + 20(27)	d2k+1e2^2k+1^-,
fcez	fcez
these two equations imply that
0(7)	-	20(27) 52 ^+ne2”(2fe+nb
+20(27) £ d2k+ne2*i(2k+n)\ Vn € Z.
fcez
We can rewrite this as
10(7/2)е-2”^2 = 0(7) £c2fc+ne2^ + 0(7) ^d2k+ne3^, kez	kez
or, in operator notation (see Lemma 7.2.1),
-^D-1 (E_n0) = 52 с2к+пЕкф + 52 d2k+nEk^. (13.25) kez	kez
298
13. Frame Multiresolution Analysis
Using the commutator relations for the Fourier transformation and the operators Ek,D,
±=П~1(Е_пф) = ^D-^Тпф = ^ПТпф-
thus, applying the inverse Fourier transform to (13.25) gives that
DTn<p = С2к+пТ-кФ 4- dik+nT-k^- (13.26) fcez	fcez
Note that the two terms on the right-hand side are orthogonal: the first belongs to Vo, while the second belongs to Wo by Lemma 13.4.2. Now let f G Wo. Since Wo C VJ and {DTf^}kez is a frame for VI, we can for each given 6 > 0 find a finite sequence {bn}nEjr such that
2
E bnDTnv - f <€.
n^J-
Via (13.26) this implies that
2
^E 6nE С2к+пТ-кф\
2
nEF k^Z
Thus also
2
E bn E d2k+nT-k^> - f < e, ne^ kez
and we conclude that span{Tk^}kez is dense in Wo- Since we already know that {7a:'0}a:Gz is a frame sequence, it follows that {T^}kez is a frame for Wo.	□
We now need to find conditions such that the three equations in Proposition 13.4.3 can be solved. It turns out that the set
Г := {7 e T : Ф(27) = О, Ф(7) > 0, Ф(7 4- 1/2) > 0}	(13.27)
will play the key role. Recall that if A is a lower frame bound for {Ткф}к&, Theorem 7.2.3 shows that for all 7 E Г,
Ф(7) > А, Ф(74-1/2) > A.
In case Г has positive Lebesgue measure we can define some functions in Wq via their Fourier transforms:
Lemma 13.4.4 Assume that ф 6 L2(K) generates a frame multiresolution analysis with two-scale symbol Ho- Assume that the set Г in (13.27) has positive Lebesgue measure and define Fi,F2 G L°°(T) by
П(?) =Xr(7), ^2(7) = Xrn[0,J[(7) -Хгп[-1,о[(7), ?е[-Ц[-
13.4 Construction of frames
299
Then the functions /1, /2 defined by
=	(13.28)
belong to Wq.
proof. Lemma 13.2.1(iii) shows that fi G Vi, i = 1,2. Furthermore, they are not identically zero; in fact, if 7 G Г, then
£|Л(27 + 2^)|2 = £|Я(7 + *Ж7 + *)|2 fcez	fcez
= 1^(7)|2Ф(7)
> A,
where A is a lower frame bound for {Ткф}ке%. To prove that the functions are orthogonal to Vq = span{Tfc^}kGz5 let к G Z; then, by Lemma 13.4.2,
= 2 J [^(7)|Ф(7)|2Жг) + Г1/2(^(7)|Ф(7)|2Яо(7))]^(7)^7-
For 7 G Г, (13.4) shows that
0 = Ф(27)>|Яо(7)|2Ф(7),
which implies that #0(7) = 0. For 7 G [— |, | [\Г, we have
Fi (7) = F2(7) = 0.
It follows that
(Ь,Ткф) = 0, V/c G Z,
i-e., /1^/2 £	О
The three equations in Proposition 13.4.3 will be the key to the next step: if they have solutions F, (9i, G2 £ L°°(T), then there exists a function ф G PVo such that {Tk'i/jjkez is a frame for Wq . In contrast to the case of a classical multiresolution analysis, the equations can not always be solved:
Theorem 13.4.5 Assume that ф G L2(K) generates a frame multiresolution analysis, and let
Г = {7 G T : $(27) = 0, $(7) > 0, Ф(7 4-1/2) > 0}.
Then the following holds:
(i) If V has positive Lebesgue meassure, there does not exist a function чф G Wq such that {Tk^}kEz is a frame for PVo.
(ii) If Г has vanishing Lebesgue meassure, then there exists a function ф G PVb such that {Tk^kez is a frame for Wq, and {D3Tk^}j^z is a frame for L2(K).
300
13. Frame Multiresolution Analysis
Proof. The proof of (i) is by contradiction, so we assume that |Г| > 0 and that there exists a function ф G Wo such that {Ткф}ке% is a frame for Wo. We can now apply Lemma 13.2.1 (iii) (with j = 0) on the frame {Tfc^}fce2 for Wo and the functions Д,/2 £ Wo defined in Lemma 13.4.4; thus, we obtain the existence of functions <71,6*2 G L2(T) such that
Л(7) = СД7Ж7), г = 1,2.	(13.29)
Since ф G Wo C Vi we can apply Lemma 13.2.1 again, this time on the frame {Ткф}ке% for Vo and with j = 1; we obtain the existence of a function F G L2(T) for which
t£(27) = F(7)^(7).	(13.30)
Combining (13.29) and (13.30) gives
Л(27) = Сг(27)^(27) = G(27)F(7)^(7), i = 1,2.	(13.31)
For 7 G Г we know that Ф(7) > 0. It follows that there exists к G Z such that <^(7 4- к) 0. Via (13.31) and the definition of fi in terms of Fi in (13.28) we obtain that
Сг(2(7 4- fc))F(7 4- к)фЬ 4- к) = F^ 4- к)ф(у 4- к), which implies that
Fi(7) = G(27)F(7), 7 ЕГ, i = 1,2.	(13.32)
We will show that (13.32) leads to a contradiction. For this purpose, we first note that
7e rn]o,^ 7-^e r n ] - |,0[.
Since we have assumed that Г has positive measure, also Г П ]0,|[ has positive measure. Let 7 G Г A ]0, |[. Then Fi(7) = Fi(7 4-1/2) = 1, so by (13.32) and the 1-periodicity of Ci we have the equations
G(27)F(7) = 1, Ci(27)F(7 4-1/2) = 1.
It follows that
о / r(7) = f(7 +1/2), 7 e r n ]o, i[.	(13.33)
We now show that a different result is obtained by looking at the function F2. If 7 G Г A [0, l/2[, then F2(7) = 1,F2(7 - 1/2) = -1. So again via (13.32),
C2(27)F(7) = 1, C2(27)F(7 - 1/2) = -1.
Adding those equations gives
^(7) = ~Fh + 1/2), 7 ё Г П ]0,1[.	(13.34)
13.4 Construction of frames 301
The equations (13.33) and (13.34) give a contradiction. Thus, if |Г| > 0, there does not exist a function G Жо such that {Tki/j}kez is a frame for JVO. This concludes the proof of (i).
For the proof of (ii) we now assume that Г is a null-set. We want to use Proposition 13.4.3 and find bounded 1-periodic functions F, Go and (9i such that the equations (13.11), (13.12) and (13.13) are satisfied. For simplicity of the formulation, we only define the functions on ТГ, with the understanding that we extend them periodically. For convenience we restate the three key equations here:
ЖРФ 4- Т1/2(ЯоГФ) = 0,	(13.11)
Я0<70Ф + ВС?1Ф = Ф,	(13.12)
Т1/2(Я0Ф)б?0 4-	= 0.	(13.13)
We split the set T into four sets which we examine separately, as we already did in the proof of Proposition 13.4.3:
T1	:=	{7	G T	:	$(7)	= 0,	т1/2ф(7)	= 0}.
T2	:=	{7	G T	:	$(7)	> 0,	Т1/2Ф(7)	> 0}.
Тз	:=	{7	6 т	•'	$(?)	> °,	Г1/2Ф(7)	= 0}.
Т4	:=	{7	G Т	:	Ф(7)	= 0,	Т1/2Ф(7)	> 0}.
In the entire proof we ignore null-sets. When needed, we let as usual A, В denote frame bounds for {Ткф}ке%- First we consider 7 G Ti. Then the equations (13.11), (13.12) and (13.13) hold for all choices of F, Gq and Gi, so we can define them to be arbitrary bounded functions on TTi; in particular we can let
F(?) = Go(7) = Gi(7) = 0, 7 C Tp	(13.35)
Now, let 7 E T2. Since Г is a null-set, we have Ф(27)	0, which by
Theorem 7.2.3 implies that Ф(27) > A. Therefore, via (13.4),
A < Ф(27)
= |Н0(7)|2Ф(7) + |Я0(7 + 1/2)|2Ф(7 + 1/2)
< (|Я0(7)|2 + |Я0(7 + 1/2)|2) В,
and
|Яо(7)|2 + |Яо(7 + 1/2)|2
< 1 (|Я0(7)|2Ф(7) + |Яо(7 + 1/2)|2Ф(7 + 1/2))
Ф(27)
А В А'
302
13. Frame Multiresolution Analysis
Altogether this shows that
л	В
-<|Яо(7)|2 +	|Но(7 + 1/2)|2<	7.	(13.36)
JD	Л	'
In order to reduce the number of unknown functions in our three equations we now define the function F on T2 by
F(7) := П/2(НоФ)(7)^-1(7). 7 £ T2.	(13.37)
As a product of bounded functions, it is clear that F is bounded. With our choice of F, the equation (13.11) is automatically satisfied. In fact, observing that
T1/2E_1(7) =	= -£-1(7),
we have
НоЯФ + Т1/2(ЖЯФ)
= ЯоТ1/2(ЛоФ)В-1Ф + П/2 (ВД/2(ЯоФ)Е-1Ф)
= ЛоП/2СЙоФ)£-1Ф- (Т1/2Яо)ЯоФЯ-1П/2Ф = 0.
The equations (13.12) and (13.13) are now two linear equations in Gq and
Gi; the determinant of the equation system is
A = Я0ФТ1/2(ГФ)-Т1/2(Я0Ф)ГФ
= Я0ФТ1/2 (Т1/2(ЯоФ)Е_!Ф) - Т1/2(ЯоФ)Т1/2(ЯоФ)Я-1Ф
= -ЯоФЯоФЯ_1Т1/2Ф - Т1/2(ЯоФ)Т1/2(ЯоФ)Я-1Ф
= -ФГ1/2Ф ()Я0|2Ф +Т1/2(|Я0|2Ф)) Е_г.
Ву (13.36) and the fact that Ф > А,7\/2Ф > A on Т2,
|Д| > Л3(|Яо|2 +Т1/2|Яо|2) >	> 0.
Thus, the set of equations (13.12) and (13.13) has a unique solution. Via Cramer’s rule,
Go
Ф F$ 0 т1/2(ЯФ)
_ ФТ1/2(ЯФ)
A	A
so
B2||F||oo ._B3l|r||
1^0(7)I S ^41	^4 Ik 11°°’
Thus Go is bounded. A similar calculation gives G\ and that also this function is bounded.
13.4 Construction of frames
303
Now let 7 € T3. Then (13.13) is automatically satisfied. In order to solve (13.11) and (13.12) we first prove that if #0(7)	0, then
У! < |Яо(7)| < У5•	(13.38)
For 7GT3 we have $(7 4-1/2) = 0, so (13.4) implies that
Ф(27) = |Яо(7)|2Ф(7).	(13-39)
Also, $(7) 7^ 0 for 7 G T3; by Theorem 7.2.3 this implies that $(7) > A on T3, so via (13.39)
Iя M|2 _ ф(27) < Я
|Яо(7)| -	-A-
This proves the right-hand inequality in (13.38). Also, (13.39) shows that if 7 G T3 and #0(7) Ф 0, then $(27) 0 0; thus
A < Ф(27) = |Я0(7)|2Ф(7) < В |Я0(7)|2,
which gives the left-hand inequality in (13.38).
Now we return to the equations (13.11) and (13.12). For 7 6 T3 they reduce to
H^F = 0, HoGo 4- FGr = 1.
In case //0(7) = 0, the first equation is satisfied, and the second equation reduces to F{y)Gi(y) = 1; this can be obtained by defining
/’(7)=G1(7) = 1.
In case 7/0(7) 7^ 0, the first equation shows that we are forced to define £(7) = 0- Therefore, the second equation simplifies to 7/0(7)Go(7) = 1, which can be obtained for a bounded function Go because of (13.38); this concludes the proof for 7 G T3. We note that one choice for the function F is
F(y\	if 7 £ ^3 and Яо(7) = 0,	4q\
| 0 if 7 e T3 and Я0(7) / 0.
The proof for 7 E T4 is similar. In this case (13.12) is automatically satisfied, and (13.11) and (13.13) reduce to
T1/2(7/oF) = 0, (Т1/2Яо)Со 4- (Т1/2^ = 0.	(13.41)
Since 7 E T4, we know that 7 - | G T3 (or, at least, its “periodic extension”). If Яо(7 — |) 7^ 0, the first equation in (13.41) forces us to define ^(7— |) = 0; this is consistent with (13.40). The second equation in (13.41) simplifies to (/1/2Яо)(7)Со(7) = 0, which is satisfied if we let Go (7) = 0. If Я0(7 — |) = 0, the first equation in (13.41) is satisfied, and the second equation gives Ti/2F(7)Gi(7) = 0; this can be obtained by letting
304
13. Frame Multiresolution Analysis
671(7) = 0. This completes the analysis of the case 7 E T4. Note in particular that no condition on ^(7) is needed for 7 E T4, except that we want F to be bounded. In particular, we can define
f(7) = 0, 7 e T4.
(13.42) □
In order to conclude that there are frame multiresolution analyses which do not lead to construction of wavelet frames for L2(K), we have to know that it is actually possible for Г to have positive measure. An example where this happens is given in [25].
In case |Г| = 0, it is worth noticing that (13.35), (13.37), (13.40) and (13.42) show how one can define F such that {Тьф}ье2 is a frame for Wo when ф is defined by 7^(27) = ^(7)^(7); in fact, we can take
Т1/2(ЯоФ)(7)Е_1(7) if 7 e TT2,
F(7) =	1	if 7 e T3 and Яо(7) = 0,	(13.43)
0	otherwise.
The proof of Theorem 13.4.5 also shows how other choices of F can be made. In particular, nothing forces us to define F on T2 as in (13.43); this choice was only made in order to simplify the calculations. Also for 7 G T3 we have a useful freedom: on the set of 7 e T3 for which #0(7) = 0, the only condition is that F(y)^1(7) = 1 for some bounded function Gj, so we can choose F to be any function which is bounded above and below on this set. In contrast, the freedom in the definition on Ti UT4 is not helpful. For 7 E Ti UT4 (or its 1-periodic extension) we have <£(7) = 0 and therefore <£(7) = 0; that is, different choices of F will not change the function ф.
Before we exploit the freedom in the choice of F further, we give an example, where we solve the three equations in Proposition 13.4.3 by direct calculations:
Example 13.4.6 We continue Example 13.2.4, and consider the function ф given by 0(7) = X[_i,i[- We have already seen that ф generates a frame multiresolution analysis. Recall that for I7I <
Я0(7) = and Ф(7) =
It is clear that Г is an empty set, so we know that we can construct a wavelet frame via the frame multiresolution analysis generated by ф. We will use the equations (13.11), (13.12) and (13.13) directly to find F E L°°(T) such that the function ф defined by
13(7) := F(7/2)0(7/2)	(13.44)
generates a frame for Wo. In our search for F we only consider 7 G [—	|[.
In order to simplify the calculations we will restrict our search to functions
13.4 Construction of frames 305
F for which
F — xi for some set I with /П	0.
о о
This simplification immediately implies that (13.11) is satisfied. (13.12) is automatically satisfied outside the support of Ф, i.e., for 7	[— |[. For
7 6 [—	|[ it reduces to
Xi-l^Go + FG^l.	(13.45)
We can satisfy (13.45) by requiring that
G0 = l on[-l,1[	(13.46)
О о and
F = GI = lon[-l -1[U[1 1[.	(13.47)
4 О О 4
We now rewrite (13.13) as
Яо(7)Ф(7)Со(7 + 1) + F(7)$(7)G! (7 + |) = 0;
using our information about F this is equivalent to
X[-t,j[(7)G!o(7 + g) + ^(7)X[-1-|[u[(7 + g) ~ °-	(13.48)
In order to satisfy this, we would like both terms to vanish. The first term will vanish if Go(7 4-	|)	= 0	on [—	|[,	i.e.,	if
( ч	n	г 1	1	lr	rl	1	lr
Go(7) = 0, 7 G -- + -[ U [- -	-[;
z	z	о	z	о	z
this choice can be made without conflict with (13.46). For the second term in (13.48) to vanish, we want F(y)Gi (7+ |) to vanish on [—|, —1[ U [|, |[; this is obtained by defining
„ r. r	1	1	lr	r1 1	lr
L	2	8	4	4 2	8l
Again, this choice is allowed. The construction gives no conditions on F on
r 1 lr rl 1 t 2’ 4 U 4’2
However, different choices of F on this set will lead to the same function ф in (13.44) because ф = and 7/1(27) = ^(7)0(7). Note that this result is in accordance with our proof of Theorem 13.4.5: in the considered example Ti = 0, and the set in (13.49) equals T4.	□
(13.49)
The choice of F in (13.43) implies that	is tight in case
{Tfct/ij/cez itself is a tight frame with frame bound equal to 1:
306
13. Frame Multiresolution Analysis
Corollary 13.4.7 Assume that ф € L2(IR) generates a frame multiresolution analysis and that {Ткф}ке2 is a tight frame with frame bound equal to 1. If |Г| =0, there exists a function тф E Wq such that {Tk^}ke% is a tight frame for Wq and {D3Tk^}j,ke% is a tight frame for L2(JR).
Proof. We first prove that {Tk^}kez is a tight frame when ф is defined via the choice of F in (13.43). By Theorem 7.2.3 it is enough to prove that the function Ф is constant outside its zero-set. Recall that
Ф(27) = |F(7) |2Ф(7) + |Г(7 + 1/2) |2Ф(7 + 1/2).
As before, we split T into the sets ТГ$,г = 1, ..,4. First we compute $(27) for 7 € T2. For 7 e T2 we have $(7) > 0 and $(7 4-1/2) > 0; since |Г| = 0 we can assume that $(27) > 0. Now,
Ф(27) = Ф(7) = Т-1/2Ф(7) = 1,
so equation (13.4) shows that
|Яо(7)|2 + |Яо(7 + 1/2)|2=1.
Thus, by (13.43),
Ф(27) = |F(7)I2 + |Г(7 + 1/2)|2 = Т1/2|Я0(7)|2 + |Я0(7)|2 - 1.
Now consider 7ET3. In this case
*(27) = |FWIW={ ;
If 7 E T4, then $(27) = |F(7 4- 1/2)|2Ф(7 4- 1/2) and 7 4- | E T3 (or its “periodic extension”); thus (13.50) shows that ^(27) only assumes the values 0 and 1. Finally, for 7 E Ti we have Ф(27) = 0.
We have now proved that Ф only assumes the values 0 and 1, so {T^jkez is a tight frame sequence; the choice of F guarantees by Proposition 13.4.3 that it is a frame for JTq.	О
As a special case we obtain the classical result already mentioned in (3.34) for construction of an orthonormal basis based on a multiresolution analysis:
Corollary 13.4.8 Assume that ф E L2(K) generates a multiresolution analysis with two-scale symbol Hq. Let F := (Tj^-^o)^-! and define the function ф E VS by 7/7(27) := F (7)0(7). Then ф generates an orthonormal basis {DjTk^}j,k<=z for L2(JR).
Proof. In case of a multiresolution analysis we have T2 = ТГ, and the proof of Corollary 13.4.7 shows that Ф = 1. Thus, by Theorem 7.2.3 the functions {Т^ф}кег constitute an orthonormal basis for PPq.	D
13.4 Construction of frames
307
Using the freedom in the choice of F we now prove that if ф generates a frame multiresolution analysis and |Г| = 0, then we can construct a tight frame {Tk^}k£i for Wo without assuming that {Ткф}к(=% itself is tight. We again refer to the splitting T = Uj=1Tj from the proof of Theorem 13.4.5.
Theorem 13.4.9 Assume that ф € L2(K) generates a frame multiresolution analysis and that |Г| = 0. Let К 6 L°°(T) be a ^-periodic function which is bounded below, and define F 6 L°°(T) by
( Т1/2(ЙоФ)(7)Е_1(7)7<(7) г/7ЕТ2;
F(7) = <j	7 € TT3 and Ho{^) = 0, (13.51)
[ 0	otherwise.
Then, with ф € Vi defined by V>(27) = F(7)0(7); the following holds:
(i) {Ткф}кЕ2 is a frame for Wq and {Б3Ткф}^к^ is a frame for L2(K).
(ii) Assume that К is chosen such that on T2 we have
1 JC|2 (ФТ1/2Ф(Г1/2(|Я0|2Ф) + |Я0|2Ф) = 1.	(13.52)
Then {Ткф}к^ is a tight frame for Wq and {D^Tk^}kez is a tight frame for L2(K); both have frame bounds equal to 1.
Proof. Compared to (13.43) we have only changed F on
Т2и{7еТ3 : Ho(7) = 0};
that is, to prove (i) it is enough to show that the three equations in Proposition 13.4.3 can be solved on this set, with the new choice of F. We have already on page 304 argued that the choice of F on T3 given in (13.51) is allowed, because Ф is bounded above and below on T3. For use in (ii), we note that with this choice,
Ф(27) = |Я(7)|2Ф(7) = 1 if 7 G Тз and Я0(7) = 0.	(13.53)
Now we check that the equations (13.11), (13.12) and (13.13) can be satisfied with the new choice of F on T2. Using that К is ^-periodic, we see that
НоГФ4-Т1/2(Яо^Ф)
= ^оТ1/2(ЙоФ)Е_!7<Ф 4- Г1/2 (^оТ1/2(^оФ)Е_1КФ)
= К (ЙоТ1/2(^оФ)Е_!Ф - (Т1/2Щ)Н^ФЕ.1Т1/2Ф) = 0.
Similarly, we can repeat the rest of the proof of Theorem 13.4.5; the fact that | К | is bounded above and below will again imply that the determinant of the set of equations determining Gq and Gi is non-zero, and that the obtained solutions are bounded (Exercise 13.2). This concludes the proof of (i). To prove (ii) we first argue that one can actually choose К such
308	13. Frame Multiresolution Analysis
that (13.52) is satisfied. Letting A, В denote frame bounds for {ТлФКег, it follows from (13.36) that
< ФГ1/2Ф (Г1/2(|Яо|2Ф) + |Я0|2Ф) < on T2. -D	/1
Since the function ФТ1/2Ф(Т1/2(|Яо|2Ф) 4- |Я0|2Ф) is ^-periodic, we can therefore choose a ^-periodic function К such that (13.52) is satisfied, and К is bounded below and above. The next step is to show that with the choice (13.52), the function Ф will only assume the values 0 and 1. The case 7 e Ti is trivial, and the case 7 € Тз,Я(7) = 0 is considered in (13.53). On T2, we get
Ф(2-) = |Г(-)|2Ф(-) + |F(- + 1/2)|2Ф(- + 1/2) = |Я|2 (Т1/2(|Я0Ф|2) Ф + |Я0Ф|2 Т1/2Ф) = |Я|2 (ФТ1/2Ф(Т1/2(|Яо|2Ф) + |Яо|2Ф) = 1.
In the rest of the cases, the function F is unchanged, so the proof 0 Corollary 13.4.7 gives the rest.	Q
13.5 Frames with two generators
In light of the fact that we can not always associate a wavelet frame {DiT^}to a frame multiresolution analysis, it is interesting to notice that we can always construct a multiwavelet frame. We need a lemma before we present the result in Theorem 13.5.2.
Lemma 13.5.1 Assume that ф G L2(K) generates a frame multiresolu^ tion analysis. For j 6 Z, let Sj : Vj -> Vj denote the frame operator for {В1Ткф}ке% and let Pj : L2(K) -> Vj denote the orthogonal projection onto Vj. Then:
(i)	For any j, к G Z, the following identities hold on Vj:
S} = DjTkS0T-kD~j and Sj1 = DiTkSo1T-kD~j.	(13.54)
(ii)	For all j, к € Z,
Р7^+1Т2к = DiTkP0D on Vo.
13.5 Frames with two generators 309
Proof. Let us fix j G Z. Then, for all к G Z and f G Vo,
SjD^Tkf = YADiTkf'DiTk'®DiT*'<l> k'ez
= Di ^(Тк/,Тк,ф)Тк,ф fc'ez
fc'GZ
= и^{/,Тк,ф)Тк,+кф k'ez
= D>TkSof.
Thus SjDJTk = DJTkS0 on Vo, and therefore Sj = DJTkSoT_kD~J on V/, the second equality in (13.54) follows from this. In order to prove (ii), we apply Proposition 5.3.5 on f G Vo and obtain that
P,D>+1Trkf = ^{^Т^З-^П.ф^П.ф-k'£z
via the commutator relation DT2k — TkD and a change of the summation index, we continue with
PjDi+1T2kf = Я^^ПР/^Т^ф^ф
= Di ^{и/,Тк,^кЗо1ф)Тк,ф
= DiTk^(Df,Tk,So4)Tk^
= DiTk^(Df,S^Tk^)Tk^
= D^TkPQDf.
□
Theorem 13.5.2 Assume that ф G L2(K) generates a frame multiresolution analysis, and let Qj denote the orthogonal projection onto Wj. Then
{DiTkQ^}jj^ U {D^TkQQDT^}jik^
is a multiwavelet frame for L2(K).
Proof. Let A, В denote frame bounds for {Ткф}ке2, and Pj be the orthogonal projection onto Vj. For each j G Z we know that {Вд+1Ткф}кЕ% is a frame for with frame bounds A, B. Since Wj is a subspace of Vj+i, it follows by Proposition 5.3.5 that {QjDj+lT^}keZ is a frame for Wj, also
310	13. Frame Multiresolution Analysis
with frame bounds A,B. Given f € L2(IR), we can write f = YljezQjf and Ц/Il2 = £VgZ ||Qj f\|2. Since Q,f € W}, we have
л|шн2 < Ei«?j/,^^+1^i2
= ^\{f,Q3D3+1T^\2
k^Z
< в ||<?л/||2.
Summing over j E Z we obtain that
A ll/ll2 < E |(/,Q^+1T^)|2 < В ll/ll2, j,fcCZ
which shows that {Q 3 Dj+1 Т^ф}	is a frame for L2(K). We now split this
family in two by considering translations with 2/c, к E Z and 2k 4- 1, к E Z separately. Observing that Qj = Pj+i — Pj, Lemma 13.5.1 implies that for any f E Vo,
Q3D3+1T2kf = {Pj+l - P3^+1T2kf
= D3+lT2kf — P3D3+1T2kf
= D3Tk(Df -P0Df)
= DjTkQ0Df-
in the last equality we used that Df = PiDf because Df E Vi. Thus, applying the result to f = ф and f — Тг ф yields
{Q3D3+1Tk<t>}3tk& = {Q3D3+1T2k<j>}3,k&U{Q3D3+1T2k+1<i>}3,keZ
— {D3TkQoD<j>}3ik<zz U {D3TkQoDT3<j>}jtk&z.
13.6 Some limitations
There are some restrictions on which frames one can obtain via frame multiresolution analysis. If	is constructed via a frame multires-
olution analysis, the orthogonal decomposition (13.8) together with the fact that {V>j,k}k€Z for a given value of j E Z is a frame for Wj imply that
,k' whenever j j', for all k,kf € Z.
For this reason the frame {ifij,k}kez is said to be semiorthogonal. Recently, Weiss et al. [232] have proposed a multiresolution analysis scheme, which also leads to a construction of wavelet frames. The obtained class contains the tight frames in Theorem 13.4.9, and the frames {^j,k}kez are not necessarily semiorthogonal. However, the freedom in the choice of the function F
13.7 Exercises
311
in the proof of Theorem 13.4.5 also makes it possible to construct non-tight frames via frame multiresolution analysis.
We have already mentioned that Proposition 7.3.6 restricts the class of functions ф which can generate an overcomplete frame	We
also note that in the case of a classical multiresolution analysis, where {2^0}fcez is assumed to be an orthonormal basis for Pq, no frame at all can be constructed. In fact, in this case we have Ф = 1 on JR, and T — T2. The proof of Proposition 13.4.3 shows that if F 6 L°°(T) satisfies the three key equations and we define ^(27) = ^(7)0(7) as usual, then Ф is bounded above and below. That is, {Tk^}kez will be a Riesz sequence, and {DjTkip}j,ke^ is a Riesz basis for L2(IR). The conclusion is that no overcomplete frame for Wo can be constructed this way, and we can not use the multiresolution analysis to obtain an overcomplete frame for L2 (R). The same happens if ф generates a frame multiresolution analysis and {Ткф}ье% is a Riesz sequence.
13.7 Exercises
13.1 Prove that the assumptions in Definition 13.1.3 are enough to make {V}, ф} a frame multiresolution analysis.
13.2 Provide the details in the proof of Theorem 13.4.9.
14
Wavelet Frames via Extension Principles
Frame multiresolution analysis is just one way to construct wavelet frames via multiscale techniques. We already mentioned in Section 13.3 that the conditions can be weakened further, and the purpose of this chapter is to show how one can still construct frames.
We will follow a fundamental idea of Ron and Shen, which (in its first version) appeared in [246]. As discussed in Section 13.3, the idea is to modify the classical multiresolution analysis definition by requiring ф to satisfy a refinement equation instead of {Ткф}к^ being an orthonormal sequence. The other conditions will be stated in the general setup in the next section; they imply that the spaces Vj defined by
Vj = &^{Ткф}ке%	(14.1)
satisfy
Vj C V3+i Vj e Z, and U~V = L2(R).	(14.2)
Thus, the multiscale idea is integrated in the setup, although it will not appear explicitly in the constructions. The multiscale feature is very important because of all its computational advances.
In contrast to frame multiresolution analysis, the purpose is no longer to construct frames for the orthogonal complement of Vj. in Vo- In fact, we will construct functions 'фг,..., фп belonging to Vi, such that the multiwavelet system {D^Tk^£}jik^z,£=i,...,n forms a frame for L2(K). In practice one usually wishes to have as few generators as possible and we show how to construct frames with two or three generators (and explain why one generator does not suffice).
314
14. Wavelet Frames via Extension Principles
After presenting the general setup, we prove the original unitary extension principle of Ron and Shen. During the last few years it has inspired a large number of authors, and we will discuss some of the most important constructions. This is followed by a discussion of more recent results, which facilitate the search for frames with prescribed properties and also lead to frames with better approximation properties. Finally, a relatively small modification of the setup leads to general (i.e., non-tight) pairs of dual wavelet frames. This construction is actually easier than its tight counterpart, and because it simultaneously delivers a wavelet frame and a dual with the same structure, it appears to be much more applicable than e.g., Theorem 11.2.3.
The various results are used to construct multiwavelet frames with B-spline generators.
14.1 The general setup
We now present the setup for the general multiresolution analysis of Ron and Shen, which enables us to construct tight frames for T2(K) of the form
U • • • U {D^Tkipn}jik^- (14.3)
As noted before, a frame of this type is called a multiwavelet frame. The functions V>i,.. •, V’n will be constructed on the basis of a function satisfying a refinement equation. Since we will work with all these functions simultaneously, it is convenient to change our previous notation slightly and denote the refinable function by v;o instead of ф. Except this, we keep the notation from Chapters 12-13; note in particular that L2(T),L°°(T) are introduced on page 284. We now list the standing assumptions and conventions for this chapter.
General setup: Let Wo E L2(R) and assume that
(i) There exists a function G L°° (ТГ) such that
= Ho(7)^o(7)-	(14.4)
(ii) lim7_+oV’o(7) = I-
Further, let Hi,., Hn G L°°(T), and define i[>i,... ,i/>n 6 £2(R) by
?M27) = ^>(7)V’o(7), € = 1,..., n.	(14.5)
14.1 The general setup
315
Finally, let H denote the (n + 1) x 2 matrix-valued function defined by
Я(7) =
( Я0(7)
Я, (7)
Т1/2Я0(7) \
Ti/2^(7)
(14-6)
\ Яп(7) Т1/2Яп(7) /
We will frequently suppress the dependence on 7 and simply speak about the matrix H.
With this setup, our purpose is to find conditions on the functions Hi,..., Hn such that 7/7,..., фп defined by (14.5) generate a multiwavelet frame for L2(JR). By Theorem 13.3.3 the spaces
Vj := span{PJTfcVio}A:GZ, j 6 Z
automatically satisfy the conditions for a multiresolution analysis in Definition 3.8.2, except (v). In Lemma 14.2.5 we prove that the general setup implies that {Tfc^ojfcez is a Bessel sequence, so by Lemma 13.3.1, -01,..., фп 6 Vi. In the literature, frames constructed on the basis of the general setup are frequently said to be MRA-based; we will not do so, because we reserve the word “multiresolution analysis” for the classical Definition 3.8.2.
Ron and Shen gave in [246] a complete chracterization of the tight frames which can be obtained via the general setup. It uses the periodization of a function f : JR -> C, which formally is defined as
я/(7) = £/(? + «)•
nEZ
If f e L^JR), then
/I  	r<x>
^2 1/(7 + n)ld7 = / l/(7)l<h < 00, hnEZ	J-°°
so £nez /(7 + n) is absolutely convergent for almost all 7 e JR. That is, P f is a well-defined 1-periodic function, and the above argument shows that PS e P(T).
Theorem 14.1.1 Let {ф^, be as in the general setup, and define the function
oo n	j — 1
©(7) := Е£|Я,(2>7)|2 П l^o(2m7)|2 j—0 £=1	m=0
with the convention П^=о |#o(2m7)|2 = 1. Then the following are equivalent:
(i) {РЗТкф^,к(=2,£=1,...,п is a tight frame.
316
14. Wavelet Frames via Extension Principles
(ii) For almost all у for which 7^(1	|2)('V) > 0, we have
lim 0(2j7) = 1 j—> — oo
and
Яо(7)Яо(7 + |)©(27) + £ Я,(7)Яг(7 + 1) = 0.
e=i
We shall not prove Theorem 14.1.1 but instead give direct proofs of some of its consequences which are especially well suited for construction of multiwavelet frames.
14.2 The unitary extension principle
The purpose of this section is to prove the unitary extension principle of Ron and Shen. It is based on the general setup in Section 14.1. We state the main result in Theorem 14.2.6, but we need some preparation first. We follow the approach by Benedetto and Treiber [25].
Lemma 14.2.1 Letg.'ipo £ L2(JR) and assume thatP^gfa) € L2(T). Then
P(gTo) = ^9’^E^E>=	<14-7)
/c£Z
and
f\ T(p^)(7)l2<h = £|(p,<MWI2-	(14.8)
•'“I	kez
Proof. Since д,фо € L2(K), we know that дфо £ ^(R), so P(^o) is well defined. Now,
(д,фоЕк)
— oo
Г\ E (^(7 + n)p(7 +	d7
_2 nez
“1 \nGZ	/
which is the /с-th Fourier coefficient for the function $2nGZ фоф + n)#(- + n). Since this function belongs to L2(T) by assumption, the lemma follows: (14.7) is just the expansion of Р(дфо) in a Fourier series, and (14.8) is Parseval’s equation.	□
14.2 The unitary extension principle 317
The first main result, proved in Theorem 14.2.6, will show that a condition on the matrix H in (14.6) implies that the multiwavelet system in (14.3) is a tight frame for L2(JR). In the proof of this, it is enough to show that the frame condition is satisfied on a dense subset of L2 (JR), cf. Lemma 5.1.7. Thus, we will in the following lemmas work with functions f € L2(IR) for which f is a continuous function with compact support, f 6 Cc(JR).
Lemma 14.2.2 Let fpo E L2(JR) and assume that lim7_>0 V’o(t) — 1- Let f G L2(IR) be any function for which f G Cc(IR) . Then, for any e > 0 there exists <7 G Z such that
(1 - e)ll/H2 < £ К/, PjT^0)|2 < (1 + e)||/||2 for all j > J.
fcez
Proof. Let j G Z,/ G L2(JR), and assume that f G Cc(JR). As a product of L2( JR)-functions, (TV f)ipQ G L^IR); thus P((7?J/)Vio) is well defined. When we only consider 7 G T, P((7?J'/)^o) can be bounded by a finite linear combination of translates of so 7?((PJ f)^o) £ L2(T). Via the Fourier transform,
(/,P>T^o) = (Ff,TDJTkiM = (PJ/,E_^o);	(14.9)
therefore Lemma 14.2.1 shows that
Ei(/,£j'7Wo>i2 = Ei^Vs-^o)!2
/cGZ	/cGZ
2
E(pJ 7(7 + п)^о(7 + n) dy.
n€Z
Now let e > 0 be given. By assumption, we can choose b G]0, l/2[ such that 1 - б < (^(t)!2 S 1 + 6 whenever |y| < b. By taking J G Z such that D3 f has support in [—5, 5] for j > J, we obtain that for all j > J,
E(£)J 7(7 + ")^о(7 + n) n£Z
2
dy ~
|(PV)(7)M7)|2d7,
and therefore
(1 - e)||^/||2 < £ K/.^nV-o)|2 < (1 + e)||^/||2-/c£Z
Since D3 and the Fourier transform are unitary operators, the lemma follows.	□
In the rest of this section we assume that {^, 77/}p=0 is as in the general setup. For a function f with f G Cc(IR), an argument as in the proof of
318
14. Wavelet Frames via Extension Principles
Lemma 14.2.2 shows that
{(/,	e £2(Z) for all £ = 1,... ,n	(14.10)
(Exercise 14.1). We can therefore define a family of functions Fj^ 6 L2(T) by the Fourier series
F^r-=^^DiT^i)E-k, j ez, £ = 0,l,---,n.	(14.11)
fcez
Since Fjj is defined in terms of which is again defined via Wo and 27^, it is natural to search for an expression for Fjj in terms of Fj# and For convenience we work with Fj-ij:
Lemma 14.2.3 Let	be as in the general setup. Then, for all
j G Z, £ = 0,1,..., n,
Fj-uh) = 2-^2(HeF3fi+T1/2(HtFjfi))^/2).
Proof. First, we use the properties of F and their commutator relations to see that
= {FD-i f,?T2kD-^t}
= (PV,E_2fcr>M
By (14.5), we can continue with
= 21/2 Г (Dij)H^E2k J — oo
= 21/2 P Р([Е^)Н^Е2к
= 21/2 If (p^Djf)TT^E2k+Tl/2P^f)7hTo)T1/2E2k)
= 21/2 If (p((D^f)HF,) + T1/2P((D^f)HF0)) E2k.
This calculation shows that (/,is the —A>th coefficient in the' Fourier expansion for the —periodic function
Р(И)<) 4- Т1/2Р((^/)Ш)
14.2 The unitary extension principle
319
with respect to the orthonormal basis {21^2E2k}kez for L2(0, |)- Using the definition of Fj-ij it follows that
Fj-M =
kez
= 2-1/2 (р((^Ш) + Т1/2Р((РТ')Ш)) (7/2). (14.12)
The function is 1-periodic, so
= НеР^Ш0).	(14.13)
Also, by the calculation in (14.9) we have (/, D^T^o) =	, Е-ьфоУ
via Lemma 14.2.1 (check the assumptions),
Fjfi = ESDjf’ E_kiME_k = Pttjyfrfo).	(14.14)
kez
Inserting (14.13) and (14.14) in the expression (14.12) for Fj-ij finally gives the result.	□
In terms of the matrix H defined in (14.6), the result in Lemma 14.2.3 shows that
/ ^-i,o(7) \ ^-1,1(7)
\ ^j'-i,n(7) /
= 2’1/2яф
(14.15)
T^F^l)
Lemma 14.2.4 Let {фе, Hyyf_Q be as in the general setup, and assume that the 2x2 matrix H^yH^y) is the identity for a.e. y. Then, for all j’EZ and all f G L2(R) for which f 6 Cc(JR),
^K/.W^^EEK/,^-1^)!2-fcez	^=0 kez
Proof. The definition of and Parseval’s equation show that
^^K/,^-1^)!2 = £ Г |f,_u|2.	(14.16)
^=0 kez	£=oJ~i
The assumption on the matrix Я(7) means that we can consider as an isometry from C2 into Cn+1 for a.e. 7 G ТГ. Using this together with
320
14. Wavelet Frames via Extension Principles
(14.15), it follows from (14.16) that
EEl<A^-1?W
£=0 k£Z
r' !-i IH> (
Fj,0(7/2) o(?/2)
2
c?7
cn+1
^,0(7/2) ^1/2^,0 (7/2)
dy c2
2'1 Г (|^,о(7/2)|2 + |Т1/2^,о(7/2)|2)</7
J-i
[\ |F,.,0(7)|2<h
2
El(/,-DWo)|2-fcez
□
Lemma 14.2.5 Let {?/^,7^}£_0 be as ^e general setup, and assume that the matrix H(7)*H(7) is the identity for a.e. 7 6 T. Then the following holds:
(i) Ы is a Bessel sequence with bound 1, i.e., Wo|2) < 1.
(ii) If f 6 L2(R), then
lim V|(/,PjT^o)|2 = 0. 7—>—00 *—* fcez
Proof. Consider a function f for which f G Cc(R). Lemma 14.2.4 shows that for any j 6 Z,
£\{f,	< £ К/, D^TM\2.	(14.17.)
fcez	fcez
Let e > 0 be given. Via Lemma 14.2.2 we can find j > 0 such that
£|(/,р^^о)|2<(1 + б)|1/112-fcez
Applying (14.17) j times shows that
El</,7Wo)|2 < ^|(/,ВЗД|2 < (1 + e)ll/ll2-keZ	k£“Z
Since e > 0 was arbitrary, it follows that
ЕКЛТН’о)!2 < ll/ll2-kez
14.2 The unitary extension principle 321
Because this inequality holds on a dense subset of L2(JR), it holds on L2(R) by Lemma 3.2.6. Thus, {Tk'ipojkez is a Bessel sequence, and the conclusion in (i) follows by Theorem 7.2.3.
For the proof of (ii), let f e L2(JR). By (i) and the fact that DJ is unitary, we know that {D^TkiMkez is a Bessel sequence with bound 1 for all j e Z. Let I C JR be any bounded interval; then
£|{/,W^o)|2 < 2£|(/Х/,^Г^о)|2
fcez	fcez
+2£|(/(l-x/),^T^o)|2
/cGZ
< 2 £ K/xz,DjTM\2 + 2 ||/(1 - X/)||2-
/cGZ
By choosing I sufficiently large, we can make ||/(1 — x/)||2 arbitrarily small. Thus, it is enough to show that
52\(fXb D3Tk^0)\2 -4 0 as j -4 -oo.
fcez
Now,
2
El(/X/,^r^o)|2 kez
2J E f f(x)^0(2ix — k)dx kez
< ii/ii22^E [\^x-k)\2dx
kezJ 1
= н/п2 E A i^owi2^.
fcGZ^2J/~fc
An application of Lebesgue’ dominated convergence theorem yields that the final expression goes to zero as j —> — oo, which concludes the proof. □
We are now ready to formulate and prove the unitary extension principle.
Theorem 14.2.6 Let {^,^}?=0 as w general setup, and assume that the matrix	is the identity for a.e. y. Then the multi-
wavelet system {D^Tk'ipe}j,kez,£=i,...,n constitutes a tight frame for L2(IR) with frame bound equal to 1.
Proof. Let e > 0 be given, and consider a function f for which f 6 Cc(IR).
By Lemma 14.2.2, we can choose J > 0 such that for all j > J,
(1 - Ю11/112 < 521</>^V-o)!2 < (1 + Oll/ll2. (14.18) kez
322
14. Wavelet Frames via Extension Principles
For any j eZ, Lemma 14.2.4 shows that
£|(/,W^o)|2 = EE
/cGZ	£=0 fcez
= £ I (/, D}~1 TWo) |2 +	E | (f, Di-1 Tk<M I2;
fcez	£=i fcez
repeating the argument on KA	it follows that for all
m < j,
Eia.^^i2 = ек/.р-г^^ + ееек/.^эд!2.
fcGZ	fcez	1-1р=тпкЕ1
Via (14.18) it follows that for all j > J and m < ;,
(l-e)IIJII2 < £|!/!TOJj|2 + fE£ia,OW2 fcez	z=ip=mfcez
< (l + e)ll/l|2-
By Lemma 14.2.5,
lim ЕКЛртГ^о>|2 =°-
fcez
Therefore, letting m -> — oo above yields that for all j > J,
(1 -e)||/||2 < E E E K/-pPW|2 < (! + e)ll/ll2-t==i p=—<x> fcez
Letting j -> oo,
n oo
(i-e)iizii2 <e E Ei(/>p₽7^>i2 < (h-oiijii2.
^=i p—-oo fcez
Since e > 0 was arbitrary, we conclude that
eeei</-p₽7^)i2=н/п2
£=i pgz fcez
for all the considered /; therefore, by Lemma 5.1.7 it holds for all f € L2(JR), which concludes the proof.	□
As noted in [25] and [112], Theorem 14.2.6 holds slightly more generally than explained here: it is enough to assume that #(7) *#(7) is the identity whenever P(|^o|2)(7) > 0.
14.3 Applications to 73-splines I
323
14.3 Applications to B-splines I
As an application of Theorem 14.2.6 we show how one can construct compactly supported tight multiwavelet frames based on splines. Note that a short survey on splines is in Section A.9. In contrast to the Battle-Lemarie wavelets discussed on page 75, the generators will be finite linear combinations of splines Bm(2x — kfi к G Z, and thus have compact support. The price to pay is that we need multiple generators.
Example 14.3.1 For any m = 1, 2,..., we consider the B-spline
V’o := B2m
of order 2m as defined in (A.14). By Corollary A.9.2,
7	f sin(7T7) \ 2m
^0(7) = —b-------- I •
\ 7Г7 /
It is clear that lim7_>0 ^0(7) = 1, and by direct calculation,
*(27) = (=< \ ^7 /
/2 sin(7T7) cos(tt7) X 2m
\ 2tt7 /
= COS2m(7T7)'0o(7)-
Thus Vto satisfies a refinement equation with two-scale symbol
#0(7) = cos2m(7T7).	(14.19)
Now, let	2J1	denote the binomial coefficient	define the
functions H1,..., Я2т £ L°°(T) by
^(7) =	sn?(7T7) cos2m-^(7Г7), z = l,..., 2m.	(14.20)
Using that cos(tt(7 — 1/2)) = sin(7T7) and sin(?r(7 — 1/2)) = — cos(tt7), it follows that the matrix H in (14.6) is given by
Я(7):	/ Ho (7)	T1/2Ho(7) \ ЯЛ7) T1/2H^) \ H2m(7) T1/2H2m(7) /
324
14. Wavelet Frames via Extension Principles
Sin(7F7) cos2rn
COS2m (7Г7)
sin2 (7Г7) cos2m 2 (7Г7)
4^7)
sin2Tn(7F7)
if 2m \ . 2m/	\	if 2m \	2m/	\
J o sm2 (777)	J 9 cos m(7F7)
\ Al \ 2m j	u	V \ 2m J	/
Now consider the 2x2 matrix M := Я(7)*Я(7). Using the binomial formula 2m / л \
(x + j/)2m = £( \xey2m-e,	(14.21)
we see that the first entry in the first row of M is 2m / о \
Miti = 52 ( 7 ) sin2£(7T7)cos2(2”l_t)(jr7) £—0	'
= (sin2(7T7) 4- cos2(7T7))2m
= 1.
A similar argument gives that M^,2 = 1. Also, using the binomial formula with x = 1, у = —1,
Mi>2
• 2m/ ч 2m/ ч Л ( 2m \ f 2тП \	f 2m \\
= sm2 (7Г7) cos2 (7Г7) ( 1 - (	!	) +	2 )-------+ \2m ))
= sin2m(7T7) cos2m(7T7)(l - l)2m
= 0.
Thus M is the identity on C2 for all 7; by Theorem 14.2.6 this implies that the 2m functions 'ipi,..., т/^т defined by
^(7) = НК?/2)^о(7/2)
_ If 2m \ sin2m+^(7T7/2) cos2rn-£(7T7/2) у \	/	(7Г7/2)2771
generate a tight multiwavelet frame {TIJTfc'0^}j>fcez,£=i,...,n for L2(R).	□
We want to study the properties of the frame constructed in Example 14.3.1, but for a reason that will become clear soon we first change the definition slightly by multiplying each of the functions He in (14.20) with:
14.3 Applications to B-splines I 325
a complex number of absolute value 1. This modification will not change the frame properties for the generated wavelet system.
Example 14.3.2 We continue Example 14.3.1, but now we define
Я^(у) =	sh/(tt7) cos2m-^(7T7), ^=l,...,2m. (14.22)
Hi only differs from the choice in (14.20) by a constant of absolute value 1, so the functions 7/7,..., ^2m given by
^(7) = Я,(7/2)^о(7/2)
=	^) sinz(7r7/2)cos2m-z(7r7/2)^o(7/2) (14.23)
also generate a tight multiwavelet frame. Instead of inserting the expression for in (14.23), we now rewrite ЯД7/2) using Euler’s formula:
„ , ,n. If 2m \ /e”7/2/e’riT'/2 + e-’riT'/2\2m~Z
H,(7/2) = ly( ( Д-----------------------) (----------------)
=	2-2my^	^^/2 _ e-7ri7/2^ ^7117/2	e-7ri7/2^2m
(14.24)
Via the binomial formula we see that (7/2) is a finite linear combination of terms
g—7117717 g— 7ri(77l —1)7	е7гг(т71 —1)7 ^7177717
All coefficients in the linear combination are real. Writing епгку = Ek/zfr) and using that
^(7) = ^ЯД7/2)Р-^о(7),
we see that is a finite linear combination with real coefficients of terms
Ek D~1zb0 = РТ-кЕ'фо — FDTk^o, к = —m,..., m. 2	2
Thus, ф^ is a finite linear combination with real coefficients of the functions
к = —m,..., m.	(14.25)
That is, ipi is a real-valued spline. Since DT^q has support in [0,m] and ЕТ_тф0 has support in [—m, 0], the spline фе has support in [—m, m]. Our arguments also show that the splines ф£ inherit other properties from ф0: they have degree 2m - 1, belong to C2m-2(IR), and have knots at Z/2.
326
14. Wavelet Frames via Extension Principles
Let us be more concrete in the case m = 1. Here, we obtain two functions Vh and First, via the expression (14.24) for Я1,
W?) = ^(772)^0(7/2)
eTviy/2 _ е-7гг7/2^е7гг7/2	e-™7/2)£,
= |(е^-е-^)П-1В2(7)
= (t_iDB2—TiDB^ (y).
Thus
(г_>ВВ2(а:) — Ti DB2(x))
= -)=(B2(2a: + 1) — B2(2a: — 1)).	(14.26)
See Figure 14.1. Similarly one proves (Exercise 14.2) that
^2(x) = 1 (B2(2x + 1) - 2B2(2x) + B2(2x - 1)),	(14.27)
which is shown in Figure 14.2.	□
We note that the computational effort increases with the order of the spline B2m we start with. For example, (14.25) shows that we need to calculate a larger number of coefficients in order to find when m increases. The number of generators ..., V>2m also increases with the order of the spline B2m- An annoying consequence is that the smoothness of the generators
14.3 Applications to .B-splines I
327
can be increased by starting with a spline B2m of higher order, but that the number of generators increases as well. In contrast, the results in Section 14.5 will allow us to construct multiwavelets with two generators which are finite linear combinations of B2m for any m G N, i.e., with any prescribed regularity.
Example 14.3.3 In continuation of Example 14.3.2 we can also construct spline frames with support on [0,2m]. We ask the reader to provide the details in Exercise 14.3. By letting V’o be the translated B-spline of order 2m given by V'o =	one can prove that
^0(27) = Но(7)^о(7)	(14.28)
with
/ 1 р-2тгг7 \ 2m
Я0(7) = f ------------ \	= e-^rny COs2m(7r7).
Since Hq appears from the corresponding function in (14.19) simply by multiplication with е-27ггш7? the functions
Я^(у) = е-27ггт7у	sin£(7T7) со82ш-г(7Г7), I = 1,..., 2m
satisfy the condition on H in the unitary extension principle. We prefer to multiply the functions with a complex number, i.e., to consider
Яг (7) = е~27гггпу	sinz(7T7) cos2771-"*(7Г7), I = 1,..., 2m;
328
14. Wavelet Frames via Extension Principles
with this choice, we conclude that the functions 0i,... ihm defined by ^г(7) = Яг(7/2)^0(7/2)
t -2nim-y If 2m \sin2m+t(7r7/2)cos2m-f(7T7/2)
V V I J	(тг-у/2)2™
generate a tight multiwavelet frame for L2(R). Furthermore, the spline functions 0i,..., 02m now have support on [0,2т]. We return to the case m = 1 in Example 14.6.2.	□
14.4 The oblique extension principle
We now return to the theoretical development. In the entire section we keep the assumptions in the general setup in Section 14.1, and our purpose is to prove a more flexible version of the unitary extension principle. Let us first give some reasons why we want to do so.
Often, it is desirable that a multiwavelet frame is generated by functions {0^}P=1 having a large number of vanishing moments. If {0£}/=1 is constructed via the general setup and the unitary extension principle, we know that 0^(7) = Я£(7/2)0о(7/2) and that 0o(O) = 1; it follows from here that the number of vanishing moments for 0£ is equal to the order of zero at 7 = 0 of Ht. This actually puts a restriction on the number of vanishing moments one can obtain for generators constructed via the unitary extension principle:	
Example 14.4.1 We return to the B-spline of order 2m considered in Example 14.3.1; it satisfies a refinement equation with two-scale symbol
Я0(7) = cos2m(7T7).
If we want to construct a frame via the unitary extension principle, the condition 7/(7) *7/(7) = I in particular implies that
1 = £|7Ш12, £=0
i.e., that
n £|^(7)|2 = l-cos2m(%7)-	(14-29)
£=1
The order of the zero at 7 = 0 for the function 1 — cos2m(7T7) is 2, so also on the left-hand side of (14.29) we can only factor 72 out; this implies that at least one of the functions |77^|2 can at most have a zero at 7 = 0 of order 2, and therefore at least one of the functions 0£ can at most have one vanishing moment.	□
14.4 The oblique extension principle 329
Another restriction on constructions via the unitary extension principle follows from Corollary 1.3.6: it shows that the assumption #(7) *#(7) = I implies that
|Яо(7)|2 + |Но(7+|)|2 <1-
Due to these restrictions certain frame constructions seem impossible when working with a set of functions {^г,#г}2=0 as in the general setup. However, sometimes another choice of these functions could lead to a surprising construction! An important reformulation of Theorem 14.2.6 was obtained by Daubechies, Han, Ron and Shen in [112]. It gives a more flexible recipe for construction of frames than Theorem 14.2.6, and is called the oblique extension principle:
Theorem 14.4.2 Let	as 9eneral setup. Assume that
there exists a strictly positive function 3 G L°°(T) for which
lim0(7) = 1
7—>0
and such that for a.e. y,
+	+ ^^(7)^(7 + ^)
£=1 _ f	ifv = Q,
~ [0	if v = |.
(14.30)
Then the functions {ОдТьф£}^ке%,£==1,...,п constitute a tight frame for L2 (R) with frame bound equal to 1.
Proof. Assume that the conditions in Theorem 14.4.2 are satisfied, and define the function t/jq G L2(R) by
^0(7) =
(14.31)
Define the 1-periodic functions Яо,..., Hn by
Яо(7) = \	Й^ = J(14.32)
V ^(7)	у t/(7)
The idea in the proof is to apply the unitary extension principle to ^о,Яо,...,Hn and thereby obtain a tight frame	>n;
finally, it turns out that	I — 1,..., n.
330
14. Wavelet Frames via Extension Principles
We now prove that ^>0, Hq, ..., Hn satisfy the conditions in the general setup. First,
^0(27) = ^/#(27)^0(27) = х/0(27)Яо(7)^о(7)
= \l~^-У 0(7)
= ffo(7)V'o(7)-
Also,
lini ^0(7) = 1™ (У0(71^о(7)) = 1-
Via the definition (14.32) and (14.30) with и = 0,
V-IH-ЫР -
I; '	- ад-|«»ы1 +Еттг
= 1,
so Яо,..., Hn е L°°(T). Because 0(2(7 4- |)) = 0(27), we also have
£я,(7)ЯД7+1) =	б(2?)	Яо(7)Яо(7+|)
г=о 2	Ve{7)0{7 + ^
+  ,	1	= Ht (7) Ht (7 + j)
y0(7)0(? + |) z=i
= 0.
Defining the functions fa,..., by
^(27) = ^(7)^(7), ^=l,...,n,	(14.33)
it follows from Theorem 14.2.6 that the functions {PJTfc'0£}J>fc6z/=i,...,n constitute a tight frame for L2(K) with frame bound equal to 1. The proof is now completed by the observation that for £ = 1,..., n,
^(27) = Яг(7)^0(7) = \/0(7)^(7) /^-г^о(7) = Й(27),
V0(7)
which shows that = fa.	□
By taking 0 = 1 in Theorem 14.4.2 we obtain Theorem 14.2.6. From the extra freedom in Theorem 14.4.2 concerning the choice of 0, one could expect it to be a more general result than Theorem 14.2.6, but the proof shows that the class of frames which can be constructed is the same for the two theorems. However, in practice Theorem 14.4.2 gives more flexibility because it naturally leads to some constructions one would not expect from Theorem 14.2.6. Let us explain this in more detail.
14.5 bewei geneiaturs
331
First, it is clear that any construction via Theorem 14.2.6 can be performed in exactly the same way via Theorem 14.4.2, simply by taking 0 = 1. On the other hand, suppose that ф0 is a compactly supported function satisfying (14.4) for some function Hq 6 L°°(T), and that the functions	= l,...,n are trigonometric polynomials satisfying the
conditions in Theorem 14.4.2. Writing ^(7) =	с^е2™*7 (a finite sum),
the definition of фе yields that
7(2?) = ВДЙ?) = ^'^смТ-к'фоЬ')-к
This shows that the frame {PJTfc'0€}j,fcGZ,£=i,...,n is generated by functions having compact support. Now, the proof of Theorem 14.4.2 shows that the same frame can be constructed via Theorem 14.2.6: we can define ф0 by (14.31), and then фе = фе with фе defined by (14.33), (14.32) and satisfying the conditions in the unitary extension principle. However, ф0 is in general not compactly supported, so the fact that the resulting frame {D^Tk^£}jtkez,£-it...tn is generated by compactly supported functions is somewhat miraculous and could certainly not be predicted in advance. In short, this shows that there are constructions which appear naturally via Theorem 14.4.2, but one would not even think about constructing them via Theorem 14.2.6.
The flexibility of the oblique extension principle is demonstrated in [112], where tight frames are obtained via some kind of interpolation between the B-splines considered in Example 14.3.1 and the functions that were used by Daubechies in her construction of orthonormal wavelet bases with compact support. We refer to [112] for details and examples.
14.5 Fewer generators
The computational effort increases with the number of generators in a multiwavelet frame, so usually we wish to have as few generators as possible. The best would be to construct a wavelet frame {DiTk^}jtke%, generated by the single function ф. However, as proved by Chui, He and Stockler [90], there are several natural cases where this is impossible:
Theorem 14.5.1 Assume that {фо,Но} are as in the general setup, and that {Ткфо}j.fcez is a Riesz sequence. If |-Ho( —1)| 7^ then there does not exist a dual wavelet frame pair {DiTkty}^^, {DjTkip}j,ke^ for which ф,ф are compactly supported and ф 6 Vi = 8рап{ВТкфо}к££-
Although there seems to be several assumptions in Theorem 14.5.1, it excludes certain desirable constructions with B-splines. Consider for example again the B-spline Bm of order m. By Lemma 3.6.10, we know that
332	14. Wavelet Frames via Extension Principles
{Bm(- — k)}ke% is a Riesz sequence, and Bm satisfies (by a calculation as in Example 14.3.1) a refinement equation with two-scale symbol
#0(7) = cosm(7T7).
In particular,
|Я0(-^)| =2-m'2.
Thus, for m 1 there does not exist dual wavelet pairs for which ip is a finite linear combination of functions
DTkBm, к E Z. However, multiwavelet frames with generators made up by finite linear combinations of DTkBm exist, as we already saw in Example 14.3.2; other constructions will be given in Example 14.6.
In passing, we note that the case m = 1 actually has to be excluded in the above discussion. In fact, the Haar function in (3.31) can be written in terms of the B-spline B\ =	namely as
^ = -)= (PT-jB, - Р^зВх) = -Er_. (PBi -	;
since the Haar function generates an orthonormal basis for L2(R), also the function -^= (DBi — PT-iBi) generates an orthonormal basis for L2(R).
Now we return to the oblique extension principle and show how to construct frames with two or three generators. We still follow the approach in [112]. Other constructions of multiwavelet frames with few generators were in fact previously given by Chui and He [88] and Petukhov [235]. They proved in particular that the general setup together with the assumption
|Яо« + |Яо(7 + 5)|2<1
always make it possible to construct a frame with two generators; if Hq is a polynomial of degree m, one can choose and H2 as polynomials of degree at most m. Petukhov also describes all solutions to the matrix equation H(y)*H(y) = I.
In order to apply the oblique extension principle one needs to choose the functions 3 and Hi,..., Hn simultaneously such that (14.30) is satisfied. It is not clear how to do this in general, but we now prove that an extra condition on the choice of 3 will make it easy to construct frames.
Corollary 14.5.2 Let ip0 and Hq be as in the general setup on page ЗЦ. Let 3 G Z°°(T) be a strictly positive function for which lim7_>o^(7) = 1, chosen such that the function
= <9(7) - <9(27) (|Bo(7)I2 + |Я0(7 + |)|2)	(14.34)
14.5 Fewer generators 333
is positive as well. Fix an integer n > 2 and let	be trigonometric
polynomials for which
£ |G,(7)I2 = 1, and £ Gz(7)Gf(7 + 5) = 0.	(14.35)
£=2	£=2
Let p,cr be 1-periodic functions such that
|p(7)|2 = 0(7), |<r(7)|2 = 77(7),	(14.36)
and define {Hi}™-} by
Я1(7) = е2^р(27)Я0(7+1),	t = 2,...,n.
Then the functions	given by (14.5) generate a tight frame
for L2(K).
Proof. We check that the functions 3 and Hi satisfy (14.30). First,
|Но(7)|^(27) + Е|Яг(7)|2
^=1
= |Ho(7)|20(27) + |Я0(7 + |)|2|p(27)P +	£ |G/(?)|2
£=2
= |Яо(7)|20(27) + |Я0(7 + |) |20(27) + ^(7)
= «(7)-
Similarly,
Яо(7)Яо(7 + 1)0(27) + £^(7)Hf(7 + 1) £=1
= Но(7)Яо(7 +1)0(27)
+p(27)p(2(7 + 1))е2-^-2^+1/2>Я0(7)Я0(7 + |)
4-<t(7)<t(7 4-1)£G£(7)GX7 4-1) ^=2
= Яо(7)Яо(7 4- 1)0(27) - 0(27)Яо(7)Яо(7 4- 1)
= 0.
□
334
14. Wavelet Frames via Extension Principles
A necessary condition for application of the oblique extension principle is that
n 0(7)H#o(7)lW7) = El^(7)|2-£=1
In particular, the expression on the left-hand side has to be positive; the condition (14.34) on p can naturally be considered as a strengthening of this. In the next section we provide examples where condition (14.34) is satisfied; as soon as this is the case, Corollary 14.5.2 makes it relatively easy to obtain frames with for example three generators. In fact, (14.35) is satisfied with
G2(7) = 7s’ Gs{7) = ?5e2”7'	(14-37)
Thus, in order to apply Corollary 14.5.2, the remaining work consists in finding such that (14.36) is satisfied. This can be done via spectral factorization, cf. Lemma 1.6.4.
The assumption (14.34) even implies that we can construct a frame generated by two functions:
Corollary 14.5.3 Let i/jq and Ho be as in the general setup on page ЗЦ. Let в 6 L°°(T) be a strictly positive function for which lim7_>o^(7) = 1, chosen such that the function p in (14.34) is positive as well. Define the functions p,a as in (14.36) and let
Я,(7) = e2-V(27)^o(7 + |), Я2(7) = Яо(7)<т(27).	(14.38)
Then the functions	given by (14.5) generate a tight frame
{DjTkipt}j,kei4=i,2 for L2(K).
The proof is similar to the proof of Corollary 14.5.2, except that one has to replace the function 3 in the oblique extension principle by 0 — p.
Note that if 3 and Ho are trigonometric polynomials, then p defined in (14.34) is also a trigonometric polynomial. The assumption that 3 and p are positive imply by Lemma 1.6.4 that we can choose p, a in (14.36) to be trigonometric polynomials. In this case, the generators in the above corollaries are finite linear combinations of functions DT^q.
14.6 Applications to B-splines II
The oblique extension principle turns out to be very useful in order to construct multiwavelet frames based on B-splines; even the extra assumptions in Section 14.5 for reduction to two or three generators can be fulfilled:
14.6 Applications to ^-splines II
335
Theorem 14.6.1 Let Bm denote the spline of order m G N with two-scale symbol = cosm(7T7). Then there exists a strictly positive trigonometric polynomial 6 for which #(7) = 1 and such that the function r) in (14.34) is positive.
Theorem 14.6.1 is proved in [112]. Thus, we can apply the results in Section 14.5 to construct multiwavelet frames with two or more generators based on any B-spline Bm. Let us for convenience consider Corollary 14.5.3; the same considerations will be valid for the other results in Section 14.5. If we choose the functions p, a in (14.36) to be trigonometric polynomials, then the functions B],B2 in (14.38) are trigonometric polynomials, which implies that the associated frame generators 7/7, p;2 are finite linear combinations of functions Bm(2x — k),k 6 Z. By choosing m large enough, we can thus obtain generators belonging to any prescribed smoothness class C'iV(R). In contrast with what we obtained for applications of the unitary extension principle, the number of generators is not forced to grow with the desired smoothness.
We give an example of frame constructions via Theorem 14.2.6 and Theorem 14.4.2.
Example 14.6.2 We return to the translated B-spline in Example 14.3.3 with m = 1; that is, we consider the refinable function тр0 = TiB2 and the associated two-scale symbol
Я0(7) = (1 + e42Tt7)2 = e-2-7 СО82(Я7).
We first revisit Example 14.3.3 and then give constructions via the oblique extension principle and its corollaries.
(i)	Defining and H2 by
#1(7)	= ге-27ГПл/2 sin(7T7) cos(tt7) =-у=е-27гг718ш(27Г7)
= 2^(1 _ е-^п) 4 v
Я2(7) = —e-2,r’7sm2(7r7) = —e42”7)2,	(14.39)
it follows from Example 14.3.3 that the functions := Vq and ?/>2 defined via (14.5) generate a tight frame for L2(JR); they are obtained by translation by one of their counterparts in (14.26) and (14.27), i.e.,
= -L(B2(2x-1) -B2(2x-3)),	(14.40)
ф2(х) = 1 (B2(2x — 1) — 2B2(2x — 2) + B2(2x — 3)). (14.41)
See Figures 14.3-14.4.
336
14. Wavelet Frames via Extension Principles
(ii)	An alternative construction can be obtained via the oblique extension principle. Let
4	4 — cos(2tt7)	t
0(7) :=-------3^’	(14Л2)
In this example we keep the choice of H2 in (14.39). Thus, if we want to use the oblique extension principle, we have to choose such that the two conditions in (14.30) are satisfied; that is, we require that
|^(7)|2 = 0(7) - |Я0(7)|^(27) - |Я2(7)|2,
Я1(7)Я1(7+ 5) = -Яо(7)Яо(7+^(27)-Я2(7)Я2(7+1).
Inserting 0, Но and leads to the equations
|Я1 (7)I2 = 1(cos(2tt7) + 2)2(cos(2tt7) - l)2,
^(7)^(7 + I) = 1(cos(2tt7) 4-2)(cos(2tt7) — 2) 2 b
x(cos(2tt7) — 1)(cos(2tt7) + I).
These equations are satisfied if we let
Hi (7) = -^=(cos(27T7) 4- 2)(cos(2tt7) — 1)
v6
= -^=(cos2(2tt7) 4- cos(2tt7) — 2)
V6
= -4-(e4jr17 + e~47riy 4- 2e2,ri7 4- 2e~2”iy - 6).
4^6
Via the choice of т/q in the general setup, ^1(7) =
^(7/2)^o(7/2)
-Е(е2™7 + e-^tt + 2е™7 4- 2e~*iy - 6)^T1B2)(y/2) 4y6
-E (Bi 4- E-! 4- 2E1/2 + 2E_1/2 - 6) P-1(^T1B2)(7) 4y 3
(T3/2 + T-1/2 4- 2П + 2 - 6T1/2) 2?B2(7).
4yd
Thus
^1(7)
(B2(27 - 3) + B2(27 4- 1) 4- 2B2(27 - 2)) 4--2—(2B2(27) — 6B2(27 — 1)).
14.6 Applications to B-splines II 337
Figure 14.3. The function given by (14.40).
This function has support on [—1,2]. Instead of taking this generator, we take
’/’?*)(7) := V’lC?-!)
(14.43)
(B2(2y - 5) + 2B2(27 - 4) - 6B2(27 - 3))
+^=(2B2(27-2) + B2(27-l)),
which generate the same wavelet system and has support on [0,3]. The function is shown in Figure 14.5.
(iii)	Systematic constructions with two generators can be given via Corollary 14.5.3. We again define в by (14.42). Then, the function rj in (14.34) is
4(7)
0(7)-0(27) (|Яо(7)|2 + |Яо(7+ 1)|2)
4 — cos(2tt7) 4 — cos(4tt7) ,	4/	.	4/ .
------5---------------5------ (cos4(tt7) + cos4(tt(7 + 1/2))) О	о
2
-(8cOS4(7T7) + 1)(COS(7T7) — 1)2(cOS(7T7) + l)2. о
Since 77(7) > 0 for all 7, the conditions in Corollary 14.5.3 are satisfied, and the remaining work consists in extracting square roots p, respectively cr of 0, respectively 77.
338
Figure 14.5. The function given by (14.43).
Writing 0(7) = — j(cos(2tt7) — 4), the general procedure for spectral factorization shows that we can take
/	,	\ 1/2
p(7) = ------- (e2”7 - 4 - л/15).
\2-3(4-x/15)/
i i < Approximation orders
339
1
2pi| Ы
Concerning the square root of rj we first find a square root of 8 cos4 (7) +1. Note that
Sy2 + 1 = 8(y4 + h
О
= 8(y2 - 81/2i)(y2 - 81/2i)
— 8(y -	- 81^4et^L)(y -	- 81/,4e*^).
According to the general theory for spectral factorization we let
z± = 81?/4el* 4- у/81/2eii — 1 = 81//4ег4 4- 1 4- \/2г,
z2 = 81/4ег^г 4- 81/2ei^L — 1 = 81//4ег^" 4- 1 — У2г.
As a square root of 8 cos4 (7) 4-1, we can take
1/2
(е-2й _ 2e~iyRe(z1) 4- Ы2)(е"2^ - 2e~^Re(z2) 4- Ы2);
evaluating this function in 777 rather than 7 and multiplying with the function ^(cos^) — 1) (cos (777) 4- 1) we obtain a square root a of rj. Inserting the expressions for p and rj in Corollary 14.5.3 we again obtain a tight multiwavelet frame with two generators; we will not perform these calculations.
The above computations are rather lengthy and cumbersome, and other choices of 3 could be made such that the spectral factorization was easier. There is, however, a special reason for the choice of 3, which will be apparent from Section 14.7. We present a related construction in Example 14.9.1. □
14.7 Approximation orders
In this section we give some more reasons for constructing frames via the oblique extension principle. More information can be found in [112]. We assume again that {//^, *s 35 *n ^he general setup, and that	is a tight frame constructed via the oblique
extension principle. Based on the refinable function we let
Vj =span{D:,Tk^0}jikez-
Let Hs denote the Sobolev space defined in (A.5). The oblique extension principle and its corollaries give some freedom in the construction of tight frames, due to the different choices of 3 one can start with. However, for practical purposes the main point is which properties we can expect of the constructed frame, and it turns out that some desirable properties will restrict the class of usable functions 3 considerably. We say that фо provides approximation order s if for all f in the Sobolev space Hs(IR),
dist(/,Vj) = O(2’JS),
340
14. Wavelet Frames via Extension Principles
i.e., if there exists a constant C > 0 such that
dist(/,Vj) < 02"js, V; el
For the tight frame {Р71к^Ьлег,£=1,...,п, we know from the frame decomposition (5.7) that for all f E L2(JR),
n
/ = Е E
^=i j,kez
As an approximation of f we can thus use
n
Qjf ~ E E E^’ DjTk^D^t
£=1j<Jk€%
for a reasonably large value of J 6 Z. We say that the frame {D^Tk^£}jtkez,i-i,...tn provides approximation order s if for all f 6 7P(IR),
II/ - Qj/II = 0(2-sJ).
When speaking about “the approximation order” it is in both cases understood that we mean the largest possible order.
By Lemma 13.3.1 we know that ^i,... ,V>n € Vi, so Qjf € Vj for all J E Z; thus, the approximation order of the frame {D3Tk^i}j,kez,£=i,...,n can not exceed the approximation order of the underlying refinable function ^o- Note that in the case of a classical multiresolution analysis, where a refinable function leads to the construction of an orthonormal basis {D3Tk^}j,kez for L2(IR), the operator Qj is the orthogonal projection onto Vj and the two types of approximation orders coincide; in general they might be different.
Since every implementation has to be done with a finite collection of vectors, the approximation order of {D3Tk^t}j,kei,t=i,...,n is clearly important in applications: we want it to be as large as possible. Assume that the refinable function ip0 provides approximation order s and consider the function
oo n	j — 1
0(7) = EEi^)|2 П l^o(2m7)i2,
j=0 £=1	m=0
which appeared already in Theorem 14.1.1. One can prove that the approximation order of {D3Tkfy}j,kez,&=i,...,n is rmn(s,p), where p is the order of zero of 1 — 0|V>o|2 at the origin. It is clear that the approximation order of {^JTk'if£}jlkez,e=i,...,n only can be maximal (i.e., reach the value s) for some special functions 0.
One can prove that for given functions Яо, • • • Hn, the function 0 satisfies
0(7) = |я0(7)|2©(7) + £|яя7)12,
£=1
14.8 Construction of pairs of dual wavelet frames 341
which is one of the two conditions in the oblique extension principle if we take 6 = 0; thus, exactly this choice for 0 is very natural.
Fortunately, in the important case where the general setup is based on B-splines, there exists an appropriate choice of 0, which also leads to fulfillment of the other conditions in the oblique extension principle. In fact, there exists a unique trigonometric polynomial 0 of minimal order satisfying the conditions; for the B-spline Bq of order 2, this is exactly the function 0(7) —	we used in Example 14.6.2(ii).
The approximation order also comes in if we want the functions to have a high number of vanishing moments. In fact, if the refinable function Wo provides approximation order 5 and all the functions	have
at least m' vanishing moments, then {DiTkip£}jtkezt£=it...,n has approximation order min(s, 2m'). Thus, high approximation orders for Wq as well as {D^Tk^p^j^kez,^!,...^ forces a large number of vanishing moments.
14.8 Construction of pairs of dual wavelet frames
So far the constructions via the extension principles have concerned tight frames. However, the technique is more far-reaching, and one can actually extend the results and construct dual wavelet pairs. We cite a result from [112], which was (in a slightly different form) also obtained by Chui, He, and Stockler [90]:
Theorem 14.8.1 Let	and {Я^,^}р=0 be two sets of func-
tions, satisfying the conditions in the general setup on page ЗЦ, and such that for some C > 0 and p >
IV’ol?)!, |^о(7)| < a.e.	(14.44)
Assume that there exists a function 0 6 L°°(T) such that lim7^o^(7) = 1 and
Hob) Hob + b^Cb) + 52 Hib)Hib + b
= ( n(7)	d4-«)
[0 zf 1/ = 1v 7
Then {DjTk^£}jtkEz,e=i,...,n and {D^Tk^t}j,kez,e=i,...,n are a pair of dual wavelet frames.
In light of the fact that the canonical dual of a wavelet frame does not always have the wavelet structure, it is important that constructions based on Theorem 14.8.1 always deliver a wavelet frame and a dual with the
Extension I’unciples
wavelet structure. Furthermore, use of this result avoids the cumbersome inversion of the frame operator, which is needed to find the canonical dual.
The decay condition (14.44) is stronger than necessary, but on the other hand weak enough to be satisfied for almost all interesting constructions. To find a pair of dual frames via Theorem 14.8.1 is easier than to construct tight frames via the oblique extension principle: the function 3 is not required to be positive.
Similar to what we saw for the oblique extension principle, we can use Theorem 14.8.1 to construct frames with multiple generators. In the rest of this section we will use the following	-
Setup for construction of pairs of dual wavelet frames:
Let {т/10,Я0}, {Hq^q} be as in the general setup on page 314, and assume that (14.44) is satisfied. Let 3 6 L°°(T) be a real-valued function for which Нт7-ю0(7) = 1, and assume that the function
7)(7) := 0(7) - 0(27) (яо(7)Яо(7) + Я0(7 + 1)Я0(7 +	(14.46)
is real-valued and has a zero of order at least 2 at the origin. Choose real-valued functions 771,772 £ L°° (ТГ) such that
4(7) = 241(7)772(7), and T)i(O) = 42(0) = 0,	(14.47)
and choose two ^-periodic and real-valued functions 0], 02 such that 0(27) = 01(7)02(7)-	(14.48)
□
Let us comment on these assumptions and choices. First, the choice of ^-periodic functions in (14.48) is possible because 7 i-> 0(27) has period |.
In the construction of tight multiwavelet frames in e.g., Corollary 14.5.2 we had to perform a spectral factorization of the functions 0 and 77. The choices of the functions 771,772,0i, 02 in (14.47) and (14.48) will replace the spectral factorization: in fact, we now prove how one can construct a multiwavelet frame based on these functions. We note that in general it is much easier to find functions satisfying (14.47) and (14.48) than to perform a spectral factorization; the price we have to pay is that we in general do not obtain a tight frame.
Corollary 14.8.2 Assume the setup on page 3^2 and define	and
{Яак ъу
Я1(7)-е2^01(7)Яо(7+|),
Я2(7) = 41(7), Я3(7) = е2^41(7),
Я1(7) = е2^02(7)Яо(7+|), Я2(7) =42(7), Яз(7) = е2^'42(7).
14.8 Construction of pairs of dual wavelet frames 343
Define the associated functions	and	as	general
setup on page ЗЦ. Then {DjTk^}jtkez,£=1,2,3 and {DjTk^e}jtkeZti=i,2t3 constitute a pair of dual wavelet frames.
proof. For и — 0,
HoWWM + E Я,(7)Ж(7) ^=1
= Яо(7)ЯоЬ)0(27) + 0, (7)02(7)ЯО(7 + ^)Я0(7 + |) +
= НоЬ)Й^)0(2у) + 0(27)ЯО(7 + ^)Я0(7 + 1)
+0(7) - 0(27) (я0(7)/^) + Я0(7 + ^)Я0(7 +
- *(7).
Similarly, for и =
3	г
Яо(7)Яо(7 + -)0(27) + £ЯД7)ЯД7 + -) £=1
= Я0(7)Я0(7 + ^)0(27)
+е2’г'701(7)Яо(7 + 1)е2-(7+1/2)02(7 + 1)Яо(7)
+Ч1(7)%(7 + |) + е2’гг7е2’г*<т'+1/2)Ч1(7)Ч2(7 + 1) = 0.
□
As for the oblique extension principle, the number of generators can be reduced to two:
Corollary 14.8.3 Assume the setup on page 342 and let
#i(?) =	+ |),	Яг(7) = е2"702(7)Яо(7 + 1),
Я2(7) = m (27)Я0(7),	я2(7) = Т)2(27)ЯО(7).
Then {-D->T/.ipt}j'kez,'(-i'2 and {D^Tk^t} j,k&,1=1,2 constitute a pair of dual wavelet frames.
We have assumed the factorizations of 0(2-) and rj to be real-valued. This is not strictly necessary. However, if в,H0 and Ho are trigonometric polynomials and , 772 and , #2 are real-valued trigonometric polynomials, then the frame generators	and {^}f=1 are symmetric if the
344
14. Wavelet Frames via Extension Principles
refinable functions фо and фо are symmetric real-valued functions. Thus, the above process will lead to symmetric dual wavelet pairs when applied to even order В-splines.
Corollary 14.8.3 is related to a recent result by Daubechies and Han: they proved in [110] that based on any two refinable functions with compact support, one can construct a pair of dual wavelet frames having generators with compact support.
14.9 Applications to B-splines III
We now return to Example 14.6.2(iii), where construction of a tight frame turned out to be cumbersome.
Example 14.9.1 We give an example of a frame construction with two generators. We will base the choices of and Я1,Я2 on the same refinable function, namely a translated B-spline of order 2. That is, we take 'фо — фо = the associated two-scale symbol is
Я0(7) =	cos2(7r7).
We again take
0(7) = 4-cos(27r7);
as proved in Example 14.6.2 this leads to
2
77(7) — t(8cos4(tt7) 4-l)(cos(7T7) — 1)2(cos(tt7) + I)2. (14.49) О
If we want to apply Corollary 14.8.3, we need to find functions 771,772,^1,^2 satisfying (14.47) and (14.48). This is easy: the expression (14.49) immediately gives several choices for 771,772, for example
771(7) = |(8cos4(tt7) 4-l)(cos(7T7) - 1)(cos(tt7) + I)2,
42(7) = (cos(tt7) - 1).
Concerning 0], 02 we simply take
14.10 Exercises 345
The functions in Corollary 14.8.3 are now as follows:
Я.(7) =
= e2«7 (1-e2^)2
6
4
^(7) = е2-702(7)Яо(7 + 1)
2„,. p	+ e-4™7\ (1 - e2™7)2
“ e V 6 J 4
Я2(7) - т?1(27)Я0(7)
1/ /е2”7+ е-2”7\4
= З 8 ----------2----)
^27гг7 e~2ттгу
2	1
4
Ze2’”7 + e~2j"7	\2 (1 + e-2”7)2
Xk 2	+1
Я2(7) = %(27)Яо(7)
/е2’*7 + е~2,г*7	\ (1 + e-2™7)2
v 2	7	4
With these choices, {DjTk^}jikez,£=1,2 and	constitute a pair of dual wavelet frames.	□
14.10
Exercises
14.1
Prove (14.10) under the stated assumptions.
14.2
Derive the expression (14.27) for the function
14.3
Prove (14.28) and provide the missing details in Example 14.3.3.
14.4
Show that the B-splines B2 and B3 defined in Section A.9 are given by
B2(x)
ВзИ
1
0
| x2
—x2
IX2 0
4- Iх з4
-h
if x 6 [—1,0], if x E [0,1], otherwise,
if x if x if x
otherwise.
2’ 2J’
X
X
9
8
9
8
e e e
346	14. Wavelet Frames via Extension Principles
14.5 Prove Corollary 14.5.3.
14.6 Prove the reduction to two generators stated after Corollary 14.8.2
15
Perturbation of Frames
In applications where bases appear, the question of stability plays an important role. That is, if {A}^=1 is a basis and {gkYkLi is in some sense “close” to {fk}™-!, does it follows that {gk}™-! is also a basis? A classical result states that if {fk}^=i is a basis for a Banach space X, then a sequence {gk}™=i in X is also a basis if there exists a constant Л G]0,1[ such that
-Рк)|| < A||52cfc/fc||	(i5.i)
for all finite sequences of scalars {q}. The result is usually attributed to Paley and Wiener [231], but it can be traced back to Neumann [226]: in fact, it is an almost immediate consequence of Theorem A.5.3 with U fk = gk-
In this chapter we concentrate on frames, so the perturbation theory takes place in a Hilbert space H. Note that if	is a Riesz basis for
H and (15.1) holds for all finite sequences, then (15.1) automatically holds for all {q}^=1 e ^2(N); thus, we can consider (15.1) as a condition on the operator
oo
К : £2(N) H, K{ck}^=1 = £ c*(A - pt).	(15.2)
k=l
For this reason К is called the perturbation operator. The same philosophy applies to the results in this chapter: all theoretical results will be obtained by putting appropriate conditions on К. We begin by stating the general results, and in later sections they are applied to Gabor frames and wavelet frames.
348
15. Perturbation of Frames
15.1 A Paley-Wiener Theorem for frames
In the entire section we assume that {fk}kLi is a frame for a Hilbert space. H. We wish to find conditions on a perturbed family	which implies
that it is a frame. As a convention we denote the pre-frame operators for {fk}kLi and	by T and U respectively, i.e.,
oo	oo
T,U : £2(N) -» к T{ck}?=1 = ^ckfk, U{ck}?=1 =
k=l	fc=l
Note that T is well defined by assumption; the pre-frame operator U is at least well defined on finite sequences, but we have to prove that
is a Bessel sequence before we know that U is well defined on ^2(N). See Theorem 3.2.3.
We first present a version of the Paley-Wiener Theorem for frames. It is due to Christensen and appeared in [72].
Theorem 15.1.1 Let	be a frame for H with bounds A,B. Let
be a sequence in H and assume that there exist constants such that Л 4-	< 1 and
||1>(а-ю|| < л||£с*а||+^(£ы2) z (15.3)
for all finite scalar sequences {c^}. Then {dk}^-i Is a frame for Tl with bounds
/ / \\2 / \2
A (1 - ( A + -£= | ) , B(l + A+-£=) .
\ \ VaJJ \ VbJ
Moreover, if {fk}^=i is a Riesz basis, then	is a Riesz basis.
Proof. {fk}^-i is assumed to be a frame, so by Theorem 3.2.3, the pre-frame operator T is bounded and |\T\| < \ГВ. The condition (15.3) implies that for all finite sequences {q},
11ЕЧ
= ||—52cfe(/fc—pfc) + £ck/k||
< j|-2LCfc(A-5*)||+]|Ea'=a||
< (i + A)||EckA|| +m(£|q!2)1/2-
This calculation even holds for all {cfc}^ G ^2(N). To see this, we first have to prove that Sfcli ck9k is convergent for any given {cfc}g2_1 G ^2(N).
15.1 A Paley-Wiener Theorem for frames 349
Given n,m 6 N with n > m.
n	m
^CkQk ~ CkQk k=i	k=i
Ck9k
k—m+l
< (1 + A)
2*2
fc=T7l4-l
/ n \ '/2
+И E №j
\fc=m+l /
since	^2(N) and	is convergent, this implies that
{Sfc=i ск9ь}п=1 is a Cauchy sequence in H and therefore convergent. Thus the pre-frame operator U is well defined on £2(N); it follows that for all
wr=i e m
oo ^скдк к=1
< (1 + A)
OO
E^A
/c=l
(15.4)
(1+А+-Я
In terms of the operators T, U, (15.4) states that
/ °°	\ 1/2
|№}£dl < (1 + A) ||T{C4^=1||+M Th
\fc=l	/
/00	\ V2
< ((1 + А)ч/В + М) ЕЫ2 , V{q}£16^(N). \fc=l	/
Via Theorem 3.2.3 this estimate shows that {^}^=1 is a Bessel sequence with bound
((1 + А)/В + м)2 = В
Now we prove that {^}^=1 has a lower frame bound. Since {A}^=1 is a frame, the frame operator S = TT* is invertible by Lemma 5.1.5, and we can define an operator
Tf : H -» £2(N), T^f := T'tTT*)-1/ = {(/, (TT*)-1^)}^. (15.5)
Note that {(TT*)-1 fk}^! is the dual frame of {A}^T1, so by Lemma 5.1.5, oo
IT/II2 = Ек/,(гт*)-1а>12
k—1
< b|/||2, V/ G H.
/1
Since Lfcii ck fk and	с^дк are convergent for all {q}^=1 e ^2(N), the
inequality (15.3) holds for all {ckJ-jEj G ^2(N). In terms of the operators T
350
15. Perturbation of Frames
and U, / OO	\
цт{С^=1-с/{Ск}г=1||<Ацт}£111+м Еы2 , (15.6) \fc=l	/
for all {q}^=1 G £2(N). Note that for f G H,
TT^f = TT*(TT*)-1/=/,
UT'f = ^(T^f)kgk = ^{f,(TT*)-lfk}9k. k—1	k=l
Using (15.6) on the sequence {q}^_1 = T^f yields
\\f-UT^f\\ < A H/Ц 4-м ll^/ll
< (a + 4=) ц/ll- v/ew.
\ v AJ
Since we have assumed that Л 4-	< 1, this implies that the operator
UT^ is invertible, and (Exercise 15.1)
Ul'T’ll < 1 + Л + -7,.	----v. (15.7)
Л	'-(А+л)
Now, f G H can be written as
f = UT^UT'y'f =	(TT"T'fk)gk-,
k=l
inserting this in the first entry of (/, f) leads to ll/ll4 = l(/,/>l2 oo	2
k=l
<	Ек([7р)-1/,(тт*)“1А)12 ElW)l2
k=l	k=l
<	J ||(t/Tt)-1/||2 E|(5b/)|2 k=l )2
ll/ll2 El(W)|2,
So
Е|(9ь/)|2>а(1-(а+^))2||/||2, k=i	' vA//
1
~ A
15.1 A Paley—Wiener Theorem for frames
351
i.e.,	is a frame for H.
For the rest of the proof we now assume that	is a Riesz basis.
To prove that {gk}^=1 is a Riesz basis we use Theorem 6.1.1 and assume that скдк = 0 for some coefficients {q}^_1 G £2(N). By Theorem 5.4.1 the lower frame bound for {fk}kL± a^so a l°wer Riesz basis bound, so (15.6) implies that
oo
fa
k=l
oo
У2Ck fa k-i
< A
/ oo	\ 1/2
+ M I 52|ck|2 I \fc=l	/
oo
Ckfk
k=l
Since Л 4-	< 1 it follows that ckfk = 0- Using Theorem 6.1.1 on
the Riesz basis {fk}kLi, we conclude that q = 0 for all к G N; therefore is a Riesz basis.	□
We illustrate Theorem 15.1.1 by an example in a general Hilbert space.
In particular, it will show that the condition A 4-	< 1 can not be relaxed.
Example 15.1.2 Let {efc}^ be an orthonormal basis for H. Given a sequence	of complex numbers, we consider the family of vectors
defined by
дк = ek + akek+1, к G N.
Then, for all finite scalar sequences {q},
11	ck (gk ~ efe) 11 — || ckakek+l | j
= (Eiw)1'1
< sup|afc[ (£|«|2) Z 
Thus, if a := supfc \ak| < 1, Theorem 15.1.1 shows that {gkYkLi is a frame (in fact, a Riesz basis) with bounds (1 — a)2, (1 + a)2.
By taking ak = 1 for all к G N, we obtain the family
9k=ek+ ek+1 к G N,
which was considered in Example 5.4.6. In particular, we know that
is not a frame. For any sequence	G £2(N),
||E«(s. -«.)|| = | |E «>+ 11 = ||EM - (Ei«i2F-
Thus, the condition (15.3) is satisfied with (Л, /i) = (1,0), or (Л, /i) = (0,1); in either case, it shows that the condition Л +	< 1 is necessary for
Theorem 15.1.1 to hold in this particular case.	□
352
15. Perturbation of Frames
The operator defined in (15.5) is the pseudo-inverse of T, cf. Theorem 5.4.3. ] stressing this point it is possible to prove a more general result than Theorem 15.1.1, where the condition (15.3) is replaced by a more “symmetric” version which also involves ^2 ck9k on the right-hand side; the exact condition is
||E CkUk - 9k) || _ *||E Ckfk I^HeMMEn2)17 ’
where Л 4-	< 1 and 7 G [0,1[. The conclusion is again that {gk}^-! is
a frame, but now the bounds also involve 7. One can actually construct examples where this condition i” satisfied, but where the condition (15.3) is not satisfied. This extension Gi Theorem 15.1.1 is remarkable in light of Example 15.1.2, which showed that one can not extend the range of the parameters Л,/1. We refer to [50], where Casazza and Christensen derive the extension as a consequence of the following remarkable generalization of Neumann’s Theorem, due to Hilding:
Lemma 15.1.3 Let U : H -» H be a bounded operator, and assume that there exist constants Л, p 6 [0,1[ for which
||t/x-x|| < A \\Ux\\4- /1 ||z||, Vzeft.
Then U is invertible.
The fact that a perturbation (in the sense of Theorem 15.1.1) of a Riesz basis is again a Riesz basis makes it plausible that if {fk}kLi is a near-Riesz basis, then a family	satisfying (15.3) is a near-Riesz basis
having the same excess. A proof of this fact can be found in [47]. Based on this result, one could easily believe that a perturbation of any frame containing a Riesz basis would again contain a Riesz basis, but this turns out to be wrong. Since this is a surprising result and forces us to deal with perturbations with great care, we present an example from [47].
Example 15.1.4 Let {ek be an orthonormal basis for a Hilbert space /С, and consider the Hilbert space TL constructed in the proof of Theorem 6.4.2, together with the frame	Recall that
1 n
fk = eLn^UL+k~ ~ Z2e(n~21)n+J’ 1 “ к “ П’
fn Jn+l
15.1 A Paley-Wiener Theorem for frames 353
Given e > 0, define the sequence	by
1 € **
J=1
1 n
9n+i =
J=1
Now, given a finite scalar sequence	we have
OO n + 1
n=l к=1
By the proof of Theorem 6.4.2 we know that {/£	a tight frame
for 7Y with frame bound 1. If we choose e < 1, then the perturbation condition in Theorem 15.1.1 is satisfied with Л = 0,/r = c, implying that is a frame for H with bounds (1 — с)2,(1 4- c)2. It contains the subfamily	n=1, which is a Riesz basis. To see this, note that
(Exercise 15.2)
^{<Ж>П==1 = h.
(15.9)
Furthermore, consider an arbitrary finite sequence {c£}, and observe that via the opposite triangle inequality and the calculation leading to (15.8),
-(1-6
354
15. Perturbation of Frames
Thus {pk}fc^i,n=i is a Riesz basis by Theorem 3.6.6. So actually we have an example where	does not contain a Riesz basis but the
perturbed family does. The opposite situation is also possible. In fact, since {9k }fcii,n=i has l°wer frame bound (1 — c)2, we can by (15.8) consider {fk Kii.’Xi a perturbation of	if < 1, i.e., if e < 1
So we get our example by choosing e < 1/2 and switching the roles of {/?)X and	□
An important special case of Theorem 15.1.1 is given by
Corollary 15.1.5 Let {А}£1 be a frame for H with bounds A,B, and let be a sequence inH. If there exists a constant R < A such that
E К/, fk - 9k)\2 < R ll/ll2, V/ e H,	(15.10)
fc=l
then {pfcjfc?-! is a frame for H with bounds /	i—\	/	i—\
a (/ - /I) ’в (/+У1) •
If {fk}kLi is a Riesz basis, then {^}^i is a Riesz basis.
Proof. The condition (15.10) corresponds to the condition in Theorem 15.1.1 with Л = 0, /1 = VRj just formulated in terms of the adjoint of the pre-frame operator instead of the pre-frame operator itself. However, an easier way to prove the frame part is to apply the triangle inequality in £2 to the sequence
{(/,№)}^! = {(/,А)}Г=1 - {{f,fk-9k)}^-	□
15.2 Compact perturbation
Another type of condition on the perturbation operator appeared in the paper [80] by Christensen and Heil:
Theorem 15.2.1 Let {fk}^! be a frame for H, and let	be a
sequence in H. If
oo
К : £2(N) К ~ EC*(A -Pk)
fc=l
is a well-defined compact operator, then	is a frame sequence.
Proof. Since {fk}kLi is a frame and the perturbation operator is assumed to be bounded, the pre-frame operator U for	is well defined and
bounded, and
1|С|| = ||т-л<цт|| + 11Л-
By Theorem 3.2.3 this implies that	is a Bessel sequence. The frame
operator for {gk}™=! is given by
UU* = (T - K)(T - K)* = S - TK* - KT* 4- KK*, where S = TT* is the frame operator for {А}£ТГ Since S is invertible, we can write
UU* = S (I + S~l(-TK* - KT* 4- KK*)) .	(15.11)
Using Lemma A.6.2 we see that 3~г(ТК* - KT* 4- KK*) is a compact operator, and that I 4- S~1(TK* - KT* 4- KK*) has closed range. By (15.11) also UU* has closed range. Since Ku = Kuu* (Exercise 15.3) we conclude by Corollary 5.5.2 that {gk}™-! is a frame sequence.	□
Note that Theorem 15.2.1 only claims that {gk}™-! is a frame sequence, i.e., it might not span the entire Hilbert space. An example where К is compact and {gk}™=! only spans a subspace is obtained by letting {A}^i be an orthonormal basis for H and
{9k}™=! = {0,А,/з, A,---}-
Perturbation via a compact operator as in Theorem 15.2.1 preserves the excess: if {fkYjfi-! contains a Riesz basis, then a total family {gk}™-! satisfying the compactness condition also contains a Riesz basis, and the two frames have the same excess (finite or not). This is proved by Casazza and Christensen in [47].
An extreme case of “perturbing” an element A in a frame	is
to replace A by zero. We have already in Theorem 5.4.7 seen that either {fk}k^e is still a frame for H, or {fk}k^£ is no longer complete. As a consequence of Theorem 15.2.1 we now prove that we still have a frame for the closed span of the remaining elements:
Corollary 15.2.2 Let {fkYjfi-! be a frame for К and {gk}™-! a sequence in H. If gk = fk except for a finite set of k 6 N, then {gk}™-! is a frame sequence.
Proof. Suppose that gk = fk except for к 6 7, where I is a finite subset of N. Then the operator
oo
McfcJtei = ТСк^к ~gk^=	-g^
к-i	kei
has a finite-dimensional range, and is thus compact. We conclude by Theorem 15.2.1 that {gk}™-! is a frame sequence.	□
356
15. Perturbation of Frames
Example 15.2.3 Consider again the family {e^ + ek±i }^=x from Example 5.4.6. {ek 4- e/c+i}^! is not a frame for span{efc 4-	= 74, and
Corollary 15.2.2 shows that {e^ 4-efc+iJjg^ can not be extended to a frame
for H by adding a finite number of elements.
15.3	Perturbation of frame sequences
In Theorem 15.1.1 we assumed that {A}^=i was a frame for the entire Hilbert space 74, and this is actually an essential assumption. If {A}^ only spans a subspace of H, a perturbation {gk}^=i might not belong to this subspace, and we can not conclude anything based on the inequality (15.3):
Example 15	.3.1 Let	be an orthonormal basis for 74 and define
the sequence
{Л}“=1 = {61,62,0,0,.
Then {fk}kLi is a frame sequence with bounds A = В = 1. Now let e > 0 be given, and consider the sequence
{дк}^ - {ei,e2, -e3, -e4, -e5,..., -ek, • • • }•
For any sequence {с^}^=1 e 42(N),
^CkUk-gk) =	-|(ElCfcl2
Thus, we can satisfy (15.3) with Л = 0 and an arbitrarily small value of /1.
However,	is not a frame for any e > 0.	□
This situation does not occur if {fk}kLi is a Riesz sequence:
Theorem 15	.3.2 Let {A}£Li be a Riesz sequence in a Hilbert space H, with bounds A,B. Let	be a sequence in H and assume that there
exist constants X,p, > 0 such that Л 4- -4? < 1 and
||E ck {fk 9k) || < A ||E	(Еы2)1/2	(15-12)
for all finite scalar sequences {c^}. Then {gk}^-i is a Riesz sequence with bounds
15.3 Perturbation of frame sequences 357
Proof. We ask the reader to prove that {дк}™^ is a Bessel sequence (check the proof of Theorem 15.1.1). Now let {c/J be an arbitrary finite scalar sequence. Then the opposite triangle inequality together with the assumption (15.12) implies that
||Е«|| - ||Гс‘Л||-||£е‘(л-й,11
> ((1 - A)/4-у) (El'tl’J 7
= л(1-(а+А))(£ы=)‘Е
□
By involving the gap (a notion introduced by Kato [196]) between certain subspaces of H one can obtain versions of Theorem 15.1.1 which apply to frame sequences. Given two arbitrary non-empty subspaces V,W of the gap from V to W is defined by
6(V, W) = sup dist(z,W) = sup inf ||z — г/||. хе У,||я|| = 1	xGV,||x||=l y^W
Let {fk}kLi be a frame sequence. As before, let T and U denote the preframe operators corresponding to {fk}(£=1 and {gk}^=1- Involving the gap between the kernels of T and U, it turns out that (15.12) is sufficient for {^}^! being a frame sequence if
6 (N’t , Nu) < 1 and A H—7=---^7-------— < 1.
y/A(l-5(NT,Nu)2)1/2
We refer to [67] for the proof. Another sufficient condition, now in terms of the gap between span {gk}^=i and spanf/fc}^! = Нт, is given by
X + -^=<	- 6(spSa{gk}™=l ,span{A}^i)2.
This version is proved in [76].
It is known that it generally is very hard to calculate the gap. This is the reason that we do not go into more detail with these results. It would be interesting to have general results that were easier to apply.
We mention a special case where the condition (15.12) applies without a bound on (A,ju) involving the gap. Suppose that {fk}^-i is a frame sequence for which the pre-frame operator T has an index, i.e.,
either dim(A/r) < 00 or codim(T^Gr) := dim(^) < 00.	(15.13)
Recall that dim(A/r) < 00 means that {fk}^-i is a near-Riesz basis for its closed span, and that dim(A/r) measures the excess. In the case (15.13),
358
15. Perturbation, of Frames
the index of T is defined as
ind(T) := dim (Л/т) — codim(T^r)-
Then it is proved in [67] that a sequence {gk}™-i satisfying (15.12) with A 4-	< 1 also is a frame sequence, and that the corresponding pre-frame
operator U has an index; in fact,
dim(A/V) < dim(A/r), codim(T^t7) < codim(T^T), and ind(/7) = ind(T).
The relation between the various dimensions is particulary interesting in the case where T is a Fredholm operator, meaning that both dim (Л/т) and codim(T^t) are finite. In this case we see that a perturbation can increase the dimension of the spanned space, but the excess will decrease with the same amount. This can be illustrated by an example in JR3:
Example 15.3.3 Let {e4i=i be an orthonormal basis for JR3 and let
= {ei,0,0}, {gi}^! = {ei, -e2,0}.
{fi}i=i spans a one-dimensional subspace, and the excess is 2. {<a}3=1 is a perturbation of in the sense that (15.12) is satisfied with (A,ju) = (0,1/2) ; however, spans a 2-dimensional subspace, and the excess is 1.	□
15.4	Perturbation of Gabor frames
In this section we return to Gabor frames {EmbTna^}m>nGz for L2(1R). There are several important perturbation questions related to a Gabor frame; we will deal with three of them, namely:
(i)	If {EmbTnag}minez is a Gabor frame and h G L2(JR) is “close” to g, does it follows that {ЕтьТпаН}т,пе% is a frame?
(ii)	If {ЕшЬТпад}is a Gabor frame and the points {(рт,п•> Атп,п)}m,nez are “close” to {(na, mb)}m>nGz, does it follows that {E\m nT^m^g}m,nez is a frame?
(iii)	If {EmbTnag}mine% is a Gabor frame and (a',&') is “close” to (a, 6), does it follow that {Emb>Tna> g}m,nez is a frame?
In all three cases we have to specify what “close” should mean. We begin with (i). One could expect {ЕтьТпаН}т>пе% to be a frame if ||# — h\\ is sufficiently small, but a result of this type turns out not to hold. Consider for example the orthonormal basis {EmTnX[o,i]}m,nGZ from Example 3.7.2; no matter how small we choose e > 0, the functions {EmTnX[o,i-e]}m,nGZ
15.4 Perturbation of Gabor frames
359
are not complete in L2(IR) and therefore can not form a frame for L2(IR), despite the fact that
I |X[O,1] ~ X[O,1 —e] 11 = 6
can be arbitrarily small. This shows that for the perturbation problem (i) it is not appropriate just to use ||p - h\\ as a measure for g and h being close.
A positive result can be obtained directly via Theorem 8.4.4 combined with Corollary 15.1.5:
Theorem 15.4.1 Let g,h 6 L2(JR) and a,b > 0 be given, and suppose that {EmbTnag}m,nez is a frame for L2(JR) with frame bounds A,B. If
R т sup ,E — h)(x — na)(g — h)(x — na — /c/Ь) < A, (15.14) b ®e[o,a] fcGZ

then {ЕшьТпаЬ,}т1пег is a frame for L2(JR) with bounds
—\	/	i—\ 2
’ в(1+У1) 
If {EmbTnag}m,nez is a Riesz basis for L2(JR), then {EmbTnahjm^nEZ w also a Riesz basis for L2(JR).
As a consequence of Theorem 15.4.1 the frame property is preserved under small perturbations measured in the Wiener space norm:
Corollary 15.4.2 Let g,h G L2(JR) and a,b > 0 be given, and suppose that {ЕтьТпад}т,пе% is a frame for L2(IR) with frame bounds A,B. If lb-fy|w,a < then {EmbTnah}mtnez is a frame for L2 (JR) with bounds
(I—	\ 2	/ i 	\ 2
1" V ьа^9 ~ B (1 + у	j •
Proof. Define again R by (15.14). By Lemma 8.5.1,
R<l\\g-h\\2w,a;
from here, the result now follows from Corollary 15.1.5 with R replaced by Ills - h\?w>a.	□
We now consider the problem (ii) of perturbing the lattice points {(na,тЬ)}т)Пб2- This problem was first considered by Favier and Za-lik [123] in 1995. Since then, several authors have studied the problem. Common for most of the results is that only the translations na or the modulations mb were perturbed. Finally, in 2001 Sun and Zhou [274]
360
15. Perturbation of Frames
gave conditions such that both could be perturbed simultaneously. To be more precise, they proved that reasonable conditions on g imply that if {EmbTnag}minEz is a frame and the Euclidean distance between (na,mb) and	is sufficiently small for all m,n G Z, then the irregular
Gabor system {E\mnT^m ,ng}m,nez is a frame as well. We state their result, which is formulated in terms of a function H depending on g G L2(IR),
/	\ 1/2
I — sup £ ls(x _ na)g(x ~ na - fc/6)| у0 xe[o,alnfcez	у
Theorem 15.4.3 Let g G L2(JR) be continuously differentiable and assume that there exist constants C > 0, a > 2 such that
< (1+^ж|)а-, toeR.
Define g(x) = xg(x). Leta,b > 0 be given, and assume that {ЕтЬТпад}т,п^ is a frame with bounds A,B. Let be any positive numbers for which
R := (SriH(g') + 8аЯ(р) + 64<тт)Я(д'))2 < A.
Then, for any sequence {(Mm,n, Am>n)}m>nGz C IR2 for which
\p>m,n - na\ < 7], |Am>n - mb\ < a, \/m, n G Z,
the Gabor system {E\m nT^m ng}m,nez Is a frame with frame bounds
/ I------\ 2	/ i------\ 2
a (x  ’в (x+У?) •
The conditions in Theorem 15.4.3 imply that there exists an open ball B(0, c) in JR2 centered at the origin and with radius c, such that any choice of points	^m,n)}m,nGZ with
(Mm.n, Am>n) G (na, mb) 4- B(0, c), Vm, n G Z
will lead to a frame {ЕхтпТ^т ng}m,nez- It is remarkable that all points (na, mb) are allowed to be perturbed equally. A significantly weaker conclusion can be obtained directly via Exercise 15.4, without any decay condition on g. In fact, assume that {Emb^na^}m,nez is a frame. Then, since the mapping (x,y) -» ЦЕяТ^Ц is continuous by Lemma 2.5.1, we can choose a sequence {(/^m,n,	nG2 0	mb))-m,n£2?, such that
£ \\EmbTnag - EXm^T^g\\2 < A;	(15.15)
m,n€Z
then {E\m пТ^т ng}mtnEz is a frame. However, the condition (15.15) will force that
|(na — ymtn,rnb — Am>n)| —> 0 as m, n —> oo.
15.5 Perturbation of wavelet frames 361
The statement of Theorem 15.4.3 indicates that the key to obtain reasonable perturbation results is to put the right assumptions on g. Feichtinger and Kaiblinger [134] recently provided strong support to this statement by proving the following important result, where g is assumed to belong to the Feichtinger algebra 50:
Theorem 15.4.4 Assume that g,h 6 and let a,b > 0 be given. If {EmbTnag}m,ntf is a frame, then there exists e > 0 such that -^na'a frame if
|a - a'| < e, |6 - b'| < e, ||p - h\|5o < e.
Theorem 15.4.4 is in a sense very surprising, even when we let h = g. In fact, when (a', 6')	(a, 6), the functions
x H- EmbTnag(x) = ei™nbxg(x - na)
and
X 1-4 Emb'Tna'g(x) = е2*гтпЬ Xg(x - na')
are moving far apart from each other for large values of m,n, so in a pointwise sense one can not consider {Emb>Tna> g}m,nez as a perturbation of {EmbTnag}m,ne%- The assumption g G <$o is important in order to obtain that }Emb'Tnai h}m,is nevertheless a frame when (a',6') and (a,b) are sufficiently close. To illustrate this, we can look at the function g = X[o,i]5 then {EmTng}minez is a frame, but {EmTn(1+e)#}m>nGZ is not a frame for any e > 0; it is not even complete, because the shifts Tn(1+^g do not cover the entire real axis.
15.5 Perturbation of wavelet frames
The perturbation theory for wavelet frames is less developed than its Gabor counterpart. In this short section we mention a few results, together with comments about missing results.
We leave the proof of the first result to the reader (Exercise 15.5).
Theorem 15.5.1 Let e L2(JR) and a > 1,6 > 0 be given, and assume that {а^2ф{а^х — kb)} jtkez 1S a frame with bounds A,B. If <p E L2(K) and
R := г sup V l(V; - <^)(a37)(V> - <p)(eJ7 + fc/b)| < A,
then	x — kb)} j,kez Is a frame for L2(JR) with frame bounds
(i—\	/	i—\ 2
1 - 71) ’в (x+71) •
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16
Approximation of the Inverse Frame Operator
Consider a frame {fk}(^=1 and the associated frame operator
oo
s-.н^н, Sf = '£/(f,fk)fk.
k=l
One of our main results is the frame decomposition (5.7), which states that
oo
к—I
In practice it can be very difficult (or impossible) to apply the frame decomposition directly: the reason is that H usually is an infinite-dimensional Hilbert space, which makes it hard to invert the frame operator. In case we can not find S-1 explicitly, we need to approximate S-1 (or at least approximate the frame coefficients {(/, S-1 A)}£Ti). In this chapter we present some methods for approximation that only use vectors in finitedimensional vector spaces. This has the consequence that all calculations can be done using linear algebra.
16.1 The first approach
Given a frame {fk}kLi with frame operator S, it is natural to try to approximate S-1 using finite subsets of {A}^.r Given n G N, the family {fk}k=\ is by Proposition 1.1.2 a frame for Hn := зрап{Д}£=1; denote its
366
16. Approximation of the Inverse Frame Operator
frame operator by
Sn : Hn Hn, Snf = £(/, A) A-	(16.2)
fc=l
Note that is finite-dimensional; thus, at least in principle we can find S~Y using linear algebra. Our first question is whether S“l approximates S'-1 in the sense that
(A5-’A) -» (A^A) as n -> oo, V/ G H,Vk G N. (16.3)
The question makes sense: for a given value of к 6 N, fk is in the domain Hn for S~x as soon as n > k. Our first result was proved by Christensen in [66].
Theorem 16.1.1 Let	be a frame. Then (16.3) holds if and only
if
Vj G N 3cj e IR : IIS-VjII < Vn>j.	(16.4)
Proof. First, suppose that (16.4) is satisfied. Fix j e N, and define
Фп = Sn'fj - S^f,, n>j.
We need to prove that for all f G H, {f, фп) 4 0 as n ч oo. Observe that
5/ = E(/,A)A = 5„/ + E (AA)A-	(16.5)
k=l	k—n+1
We will use this to obtain an alternative formula for фп. First, since
Зфп — SSn fj ~ fa,
an application of (16.5) on S^1 fj yields
Зфп = sns^f] + £ k—
= E О.м fc = n+1
It follows that
Фп = E	n>j.
k=n+l
16.1 The first approach 367
Therefore, for f e H,
I7.O2
oo
fc=n-H
oo	oo
<	E l(Sn7,,A)l2 E K/^-'A)!2 k—n-f-l	/c=n+l
oo
<	ВИЗЕЛИ2 E 17,5-7&>12
k—n+1
oo
<	E l^”7,A)l2-
k—n+1
Since {fkYkLi is a frame, K^-1/, A)|2 —> 0 as n -> oo. Therefore our estimate proves that (/,фп) -> 0 as n —> oo, as desired. On the other hand, if we assume that (16.3) is satisfied, we can fix an arbitrary j 6 N and consider the functionals
n>j.
Each An is bounded, and by (16.3) the family of operators {An}n>j is pointwise convergent; by Theorem A.5.1 the family of norms {||АП||}П>7 is therefore bounded, i.e., there is a constant Cj > 0 such that
|Ц„|| = 115-7,11 < Vn>j.
□
Via Proposition 5.4.4 we obtain the following immediate consequence:
Corollary 16.1.2 Assume that {fkYkLi is a Riesz frame. Then (16.3) holds.
In particular, (16.3) holds if {A}£Ti is a Riesz basis. Intuitively, one could expect the same to be true if {fk}^i is “close to be a Riesz basis”, but this turns out not to be true. After adding a single element to a Riesz basis the property (16.3) might no longer hold:
Example 16.1.3 Let {e^}^ be an orthonormal basis for ?/, and define
fi = ei, fk = ek-i 4- yefc, к > 2.
к
By Example 15.1.2 we know that {A}£T2 *s a Riesz basis with bounds |, so {fk}(^=1 is a frame with excess equal to 1. For n E N we want to find f := Sn1 fi, i.e., to solve the equation
n
^f,fk)fk = fl, fenn.
k=l
368
16. Approximation of the Inverse Frame Operator
In terms of the orthonormal basis	the equation can be written as
E (hf,fk) + (f,fk+i)}ek + -{f,fn)en=e1.	(16.6)
/ n
It follows from here that
= 0 and that </, A) = -k{f, A+i), к = 2,...,n - 1,
so = 0 for all к = 2,..., n. Again by (16.6) we have (/, A) = 1; expressing the last two conclusions in terms of {e^}^ we have
(/,ei) = l, (/,e2) = -2(/,ei> = -2,
and in general
<f,ek) =	= (—l)k-1fc!, fc = 2,...,n.
Since f 6 Hn = span{efc}^_1, this implies that
f =	<16-7)
fc=l	k—1
In particular,
/ n	\ 1/2
ll^n'All = I E(fc!)2 ) ->ooasn-Oo. \fc=l	/
Therefore (16.4) does not hold. We can actually be more concrete and exhibit a vector g 6 H for which the desired convergence in (16.3) fails; in fact, with
the expression for Sn rfi in (16.7) shows that
(g,S~1fi)=n.	□
The question of convergence in (16.3) can in fact be used to give yet another characterization of a frame being a Riesz basis:
Proposition 16.1.4 A frame {fk}kLi is a Riesz basis if and only if {fk}kLi is linearly independent and (16.3) holds.
Proof. That a Riesz basis satisfies (16.3) follows by Corollary 16.1.2. Now assume that {A}£Li is linearly independent and that (16.3) holds. Let n G N. The linear independence of {A}£Li implies that {fk}k=i a (Riesz) basis for 7Yn. By Corollary 1.1.6, the dual basis is {S^1 A}fc=i> so
= 6k,j, k,j = 1,2,...,n.
16.2 A general metnud
By letting n —> oo and using (16.3), we obtain that
=8k.j, VfcjeN.
By Theorem 6.1.1 we conclude that {fkYjfLi is a Riesz basis.	□
16.2 A general method
In this section we derive a method for approximation of the inverse frame operator which works for all frames. It can be considered as an improvement of the method from the last section. The initial results (up to Theorem 16.2.3) were proved by Casazza and Christensen in [48], and the rest are from [68]. We keep our previous notation, and let {fk}kLi denote a frame with frame operator 5; we again consider finite subfamilies {fk}k=i anc^ the associated finite-dimensional vector space Hn = зрап{Д}£=1. Let Pn denote the orthogonal projection of 77 onto 7Yn, and let
In := {1,2,..., n}, n e N.	(16.8)
With this notation, the purpose is to approximate S-1 via sets of the form {fk}kein,n £ N. We note that it is only for notational convenience that the following results are formulated for a frame indexed by N and for this choice of Zn; given a frame indexed by a countable set I, similar results with identical proofs holds for any family {In}^=1 of finite subsets of I for which
h C I2 C • • • C In f
In order to make the general result clear from our presentation, we will denote the number of elements in In by |In |, despite the fact that with our choice (16.8) we simply have \In\ = n.
We begin with a Lemma.
Lemma 16.2.1 Let	be a frame for H with lower bound A. Given
n G N, there exists a number m(n) such that
.	n+m(n)
v ll/ll2 < E l(/,A>l2, VfeHn.	(16.9)
k=l
Proof. Let n 6 N. Given e > 0, choose a finite set of elements {pj}/=1 in Hn such that ||pj|| = 1 for all j = 1,..., <7, and such that the balls
B(Pj,e) :={/GHn : ||/-p,||<e}
cover the compact set {f e Hn | ||/|| = 1}. Since
00
A < El^-A)l2, =
k=l
370
16. Approximation of the Inverse Frame Operator
we can choose m(n) such that
n+m(n)
A;< £ lto,A>|2,	=
fc=l
Now let f 6 7Yn, ll/ll = 1. Choose j such that / G B(^, e). By the opposite triangle inequality applied to
{(/> А)}^Г(П> = {to- A) - to - A A)}kir(n).
we have (letting В denote an upper frame bound for {fk}kL±)
z .	\	1/2	/	,	/ X	\	1/2
n+m(n)	\	/	n+m(n)	\
E l<AA)l2 > E lto>A)l2J
/c=l	/	\	b=l	/
/ । / \	\	V2
/ n+m(n)	\
- E lto-AA)l2
\ k=1	/
> ^7l-v^lto-/ll
By choosing e small enough, ^/a| — y/Be > from which the result follows.	□
The next lemma shows that for any frame {fk}^-^ we can construct a family of frames “approaching	which have common frame
bounds. Remember that (16.3) holds for every Riesz frame; the lemma below turns out to be the key to an improved method that works for every frame.
Lemma 16.2.2 Let {fkYjfLi be a frame with bounds A, B. For any n G N, choose m(n) such that (16.9) is satisfied. Then {Pnfk}^™^ is a frame for Hn with bounds ^,B; the associated frame operator is
-^71*^71+771(71) •
and
HPnSn+m(n)|| < B, ||(P„Sn+m(n))-1|| <
16.2 A general method 371
Proof. Fix n G N and let f G Pn- Then, with our choice of m(n),
n+m(n)	n+m(n)
E \(f,Pnh}\2 = E k/,a>i2 fc=l	fc=l
> у ll/ll2-
Also, n+m(n)	n-bm(n)	oo
E i(/.-PnA)i2= E k/,a>i2 < Ei<^)i2
k=l	k=l	k=l
< в н/п2.
So	*s a ffame for Hn with the claimed bounds. The frame
operator is given by
n+m(n)	n+m(n)
E U.Pnfk}Pnfk = Pn E </•/*>/* k=l	k=l
— Tn$n+m(n)/> f ^n-
Now where PnSn+m(n) is identified as the frame operator for a frame with bounds у, В, the norm estimates for PnSn+m(n) and (FnSn+m(n))-1 follow from Proposition 5.4.4.	□
We are now ready to prove that S-1 can be approximated arbitrarily well in the strong operator topology using the operators
(Fn'S'n+mfn))  Pn “> Pn, П G N.
Note that Pn3n+m(n) is an operator on a finite-dimensional vector space. This implies that its inverse, and therefore (FnSn+m(n))-1 Pn, in principle can be found using finite-dimensional linear algebra. In practice, of course, large values of n will complicate the calculations.
Theorem 16.2.3 Let {fk}kLi be a frame with bounds A,B. For n G N, choose m(n) such that (16.9) is satisfied. Then
(PnS^^Pnf^S^ffom^oo, yfEH.
Proof. Let f 6 P. Then
8 1 f ~ (T>n‘S’n+m(n)) iPnf = PnS 1 f — (PnSn+m(nf) 1Pnf +(z — pn)s-1/.
Since (Z — Pn)S~1f -» 0 as n -> oo, it is enough to show that
'Фп	Pn.8 f (-^п^п+тгЦп)) Pnf 0 as n > oo.
372
16. Approximation of the Inverse Frame Operator
Since фп € Hn we can apply the operator PnSn+m(n) get
Фп = (PА+т(п)) 1 {Pn^n+m(n)Pn^ f ~ Pnf)-
Consequently, via Lemma 16.2.2,
IltM < ||(ASn+m(„))-1|| \\P„Sn+m{nyPnS-'f-Pnf\\ < j \\Sn+m{n)PnS-'f-f\\
—> 0 for n —> oo.
□
At the moment the results are purely theoretical: they depend on the choice of m(n), which has not been estimated yet. We now want to obtain a more explicit result, giving more information about how to choose m(n) such that (16.9) is satisfied. We begin with
Lemma 16.2.4 Let {fk}^=1 be a sequence in H and let n G N. Let An denote a lower frame bound for the frame sequence {fk}k=r Then for any set Jn containing In,
£ К/, A>|2 < maXj^ £ |(A,/,)|2 Ц/H2, V/ e нп k$jn	n
Proof. Let / 6 Hn. Since {A}£=i is a frame for we can use the frame decomposition / = ^j&In (f,	to get
К/, Л>12
(£(/,V/,>/,,A>
j£ln
j£ln
Now, by Cauchy-Schwarz’ inequality and the fact that {Sn 1fj}jeln is a frame for Hn with upper bound we have
КЛАЛ2 < £|(/,5-1Л)|2 £|(/,,A)I2
jeln	j(=In
< А н/п2 £кл,а)12-n jeln
16.2 A general method
373
Thus
Ekwi2 <	Ej-ii/ii2	Eiu-,a>i2
kf£Jn	k$Jn n	j<=In
=	7- н/п2	E	E	кл.am2
n jeln k£Jn
<	Ibi ll/ll2	maxje;„	E l(/n/*>|2-
П	k£Jn
□
Theorem 16.2.5 Let {fk}kLi be a frame for H with bounds A,B. Let {бп}£Т1 C]0, A[ be a decreasing sequence of numbers converging to zero. For n 6 N, choose a finite set Jn containing In such that
E КЛА)12<еп11/112- V/eft„.	(16.10)
k(£Jn
Let Vn : Lin —> Hn denote the frame operator for the finite family {Pnfk}keJn- Then, for all f &H,
US-1/ - VA/Ц < л(/1бп) Н/П +	+ 0 li(/ " P")5"7||.
Proof. Let n 6 N. Denote the restriction of PnS — Vn to 3in by fiPnS — Vn)\Hn‘ the reader can check that (PnS — Vn)^n is self-adjoint. Furthermore, for f G Hn we have
{(PnS-Vn)\HJ,f) = lfPnS-Vn)f,f)
= {PnSf,f)-{Vnf,f)
= (E</, fk)Pnfk,f) - ( E (/.AAIAA,/) \fc=l	/ \keJn	I
= Ei</,a>i2- E kaa>i2
k—\	k£jn
= El</>A)l2>0.
k<£Jn
It follows from (A.8) and the condition (16.10) that
||(PnS-K)|Wn|| = sup/6Hn>||/||=1|((P„S-V„)/,/)|
= suP/ew„,||/||=i E l</> A>l2
k(£jn
374
16. Approximation of the Inverse Frame Operator
Also, for f e Hn,
El(/,PnA)|2 = Ek/,a>i2
k£Jn
oo
= Ekaaji2- E k/,a)i3 fc=l	k(£Jn
> (A-e„)||/||2.
So A — en is a lower frame bound for {Pnfk}keJn\ by Proposition 5.4.4 this implies that ||Vn-1|| <	• Now let f 6 H. We have
<	iKz-Pn^-VlI + UPnS-V-VP^H
<	11(7 - Pn)S-1/|| + IIVII \\VnPnS^f-Pnf\\
<	11(7 - P„)S-7ll + 7^— IIKPnS-1/ - Pn/||.
A — €n
Now,
||v„p„s-7-p„/|| < iiv^s-1/- PnSP„s-7ll
+\\PnSPnS~1f-Pnf\\
<	IKK - p„s)pns-1/l| + \\spns-'f - /II
<	en HPnS-'/ll + 115Ц ||PnS~7 - S"7||
<	ll/ll+ P ||(7-Pn)S-1/||. /1
Altogether,
IIS-1/ - VK/II < eE ' ll/ll + (-A- +1) П(/ -ЮЯ"1/!!.
□
It is always possible to chose a set Jn such that (16.10) is satisfied (Exercise 16.2). By Theorem 16.2.5, this choice of Jn implies that
v~lpnf ->s-1/ for n -> oo, yfe-H.
That is, the operators {V~} Pn}^=1 converge to S-1 in the strong operator topology. In particular, the frame coefficients can be approximated:
(/, v-'Pn/k) -» (/, s-1 A) for П oo, V/ e К к e N.
Compare this conclusion with the starting question (16.3): the obtained result is similar to what we originally wanted (and proved impossible without further conditions), but to find V~x is more involved than to find S~r.
Under the conditions in Theorem 16.2.5 it even holds that the sequence of coefficients {(/, V~1Pnfk)}keJn, n G N, converges to {(/, S"1in 72-sense as n —> oo:
16.2 A general method 375
Theorem 16.2.6 For n E choose Jn as in Theorem 16.2.5. Then
E \{rv-ipnfk}-u,s-4k)\2+ E \u,s-xm\2
0 for n -> oo, V/ EH.
Proof. Let f E 'H. It is clear that К/, ^-1A)|2 -> 0 for n —► oo. Concerning the first term, we have
E i(/>kt1^a)-</,s-1a)i2
k£Jn
= E \<PnS.VnlPnfk} -k€.Jn
keJn
< b||(V^-s-W
—> 0 as n ч oo.
□
Observe that since {Pnfk}keJn is a finite set, the frame operator Vn and its inverse can be found using finite-dimensional linear algebra. This does not make it trivial to apply the results, but calculation of V~1Pn is a drastic simplification compared to inversion of S.
For applications of Theorem 16.2.5 the pure existence of sets Jn satisfying (16.10) is not enough — we need to to able to find Jn. Combining Theorem 16.2.5 and Lemma 16.2.4 we are able to replace (16.10) — a condition that has to be satisfied for all f E 7Yn — by a condition only involving the finite set of vectors fj,j E In:
Theorem 16.2.7 Let {fk}^=i be a frame for H with bounds A,B. Let C]0,A[ be a decreasing sequence of numbers converging to zero.
For n E N, let An denote a lower frame bound for the frame sequence {fk}k=i and choose a finite set Jn containing In such that
E кл.л>|2< тй,	(i6.li)
k(£Jn	1 nl
Let Vn : Hn TLn denote the frame operator for the finite family {Pnfk}keJn- Then, for all / E 7Y,
115’7 - V^/II < -T7p-—. ll/ll + (-A- +1) ll(/ - Pn)$-7||. zi^zi tn)	\ zi 6n J
Observe that for n E N, (16.11) consists of \In | conditions on the set Jn. In the next section we apply this result to Gabor frames, where the number
376
16. Approximation of the Inverse Frame Operator
of conditions can be reduced further. Applications to wavelet frames are given in [68]; also in this case the number of conditions can be reduced.
16.3 Applications to Gabor frames
As noted at the begin of Section 16.2, the methods for approximation of the inverse frame operator can also be applied to frames indexed by Z2: we only have to replace the index sets	in (16.8) by finite subsets of
Z2 for which
Ii C Z2 C • • • C In t z2.
In this chapter we denote Gabor frames by {EkbTiag}k,i^i\ recall that here a, b > 0 and g 6 L2(K). For a Gabor frame it is natural to chose the sets In as “finite lattices”,
In := Ж Z) E Z2 : \kb\ < Dn, \la\ < Cn},	(16.12)
where C, D are positive constants; the freedom in the “lattice size” gained by introducing these constants will prove useful. Our purpose is to show that the condition on the finite set Jn in Theorem 16.2.7 can be simplified in the case of a Gabor frame. By taking Jn of the form Jn = In+m(n) the question is how to find a value for m(n) > 0 such that (16.11) is satisfied.
Lemma 16.3.1 Let n G N and m(n) be an arbitrary nonnegative integer. Then, for all (k ,1 ) G In we have
£ \{Ek.bTl>ag,EkbTlag)\2 <	£	|{EkbTlag,g)|2.
(fc,0if/n+o.(n)
Proof. Let (fc ,l') 6 In. Then
£ \{Ek/bTl’ag,EkbTiag)\2
(fc, Z) In Tn (n)
~ У? K^'(fc-fc/)b^r(Z-Z/)a^’^)|2
(fc,/)^/n_|_Tn(nj
= £	-	£ I ft'Z'’ ff) I2
Mez	(k,i)et„+m(„)
= £ \(EkbTlag,g)\2- £	\(E{k_k,,bT^,ag,g}\2.
k,iez	(*,
16.3 Applications to Gabor frames
377
Let C, D > 0 be the constants in (16.12); then
52	tf)|2
(fc,Z)G/n_|_Tn(n)
=	52	52
|fcb|<(n-bm(n))D |/a|<(n+m(n))C’
> E E \<EkbTlag,g}\2
|fcb|<(m(n))£> |Za|<(m(n))C’
E \(EkbTlag,g)\2.
(к, I) G Im (n j
It follows that for all (k‘, I ) G 7n,
52	\(Ek'b^'ag, EkbTiag)\2
( A., Z ) In -j- m ( n )
< E \{ЕкьТ1ад,д}\2 - E l(^TloP>P>|2
fcjGZ	(A:,Z)6/Tn(n)
E <• ,
(fc,0£Ln(n)
□
Combining Theorem 16.2.7 and Lemma 16.3.1 we get
Theorem 16.3.2 Let {EkbTiag}k,ie% be a Gabor frame with bounds A, В and let	C]0,A[ be a decreasing sequence of numbers converging to
zero. For n 6 N, let An be a lower frame bound for the frame sequence {EkbTiag}(k,i)ein and choose a number m(ri) such that
E \{EkbTlag,g)\2 <^.	(16.13)
(k,l)elm{n}	1 nl
Let Vn : Hn -> Hn be the frame operator for {PnEkbTiag}^^eln+m{n}. Then, for all f 6 L2(JR),
/ D	\
IIS-1/ - VPn/H <  n ll/ll + —— +1 11(1 - pn)s-'f\\. tn)	y/1 tn /
Thus, in the case of a Gabor frame the single condition (16.13) is enough to determine the choice of Jn. Observe, that by the frame condition /eZ \(EkbTiag,g)\2 is finite; thus, to satisfy (16.13) is “only” a question of choosing m(n) sufficiently big.
For the Gaussian a direct estimate for (fc,/)^/Tn(n) \(EkbTiag,g)\2 can be given:
378	16. Approximation of the Inverse Frame Operator
Example 16.3.3 Let g(x) = 2r/4e 7ri2. It is well known, cf. [138], that \(EkbTlag,g}\=e-^a2+‘2b^^2.
Thus, taking C = D = 1 in (16.12),
E \(EkbTlag,g}\2
(M)£Ln(n)
|A;|>m(n) /€Z

e-^’Ee~’W2
k=0
4 12 e~*ka2 E6-^2 + E
yfc=(m(n) + l)2	1=0	/=(тп(п)Ч-1)2
/ е-7гa2(m(n)+1)2 g—7гЬ2 (m(n) + l)2 \
4 1	’
(1 - e-7rb2)(l -e-™2)

16.4 Integer oversampled Gabor frames
In this section we consider a Gabor frame {EkbTiag}k,ie% which is integer oversampled, i.e., we assume that
ab =	, where N G N.
In this case we choose the index sets In in (16.12) as
Jn:={(fc,z)ez2| |fc|,|Z|<nV}.
With this choice of the index set Zn, Theorem 16.3.2 applies with
|In| =(2nV + l)2.
We will show how to obtain estimates for the approximation rate for the dual window S-1g in the case of integer oversampling. As before Pn will denote the projection of L2(R) onto
Hn = span{EfcbT/a^}(fc)/)GZn = span{EkbTiag}\k\,\i\<nN.
Let HH* be the Gram matrix for {Em/aTn/bg}m,nez, as defined in (9.13). In case g and g decay exponentially, Strohmer proved in [268] that there exist constants C, A > 0 such that
Using Lemma 9.3.3 it follows that for some C", A',
|[(ЯЯ‘)-1]1Л;О,о| <C'e-A'(l'=l+l,D.	(16.14)
16.5 The finite section method 379
Theorem 16.4.1 Suppose that {EkbTiag}k,ie% is an integer oversampled frame and that g and g decay exponentially. Under the assumptions in Theorem 16.3.2, there exist constants X,C > 0 such that
\\S~'g - V-lPng\\ <	Н/П + Ce~Xn, Vn 6 N.
tn)
Proof. By the Janssen representation (9.14) of the inverse frame operator we have
S-ig = ab
= ab [(HH^^^i^-ooEkNbTiNag-kflE.'Z
For |k\, |Z| < n, we have that PnEkNbTiNag = EkNbTiNag' Thus
(I - Pn)S~lg = ab(I - Pn) E [(HH'r^iwoEkNbT^g.
|fc|>n ОГ |/|>n
By Theorem 9.2.6 we know that {Ek/aTi/bg}kjez is a Riesz sequence with upper bound Bab. Therefore the subfamily {EkNbTiNag}^>n or||/|>n is also a Riesz sequence with upper bound Bab. Using the estimate (16.14) for [(ЯЯ‘)-1]|Л;оо, we get
Wil-Pn^gW2
<	Bab(ab)2 £	|[(ЯЯ*)-1](Л;О,о|2
|/c|>n ОГ |/|>n
<	Я(С')2(а&)3	£ e-2V(lM+HI)
|k|>n ОГ |/[>n
<	B(C")2(ab)3 | e~2V|fc| У2е-2Л^1+ У2 e~2A^ y^e~~2A |fe* j \|k|>n	zgz	\i\>n	kez	/
p~2X'
<	8B(C-)2W3^^)2e-2A-.
Now the result follows from Theorem 16.3.2.	□
16.5 The finite section method
In this section we present a direct method for approximation of the dual frame for a Gabor frame {EkbTiag}k,iez. was developed by Strohmer [268] and is much simpler than the method in Section 16.3. This is not a surprise: the method in Section 16.3 applies to an arbitrary frame and does not take the special structure of a concrete frame into consideration.
380
16. Approximation of the Inverse Frame Operator
Formula (9.14) and Theorem 9.2.6 are the main ingredients for this approach. The key point is that the approximation problem for Gabor frames can be translated into an approximation problem for Gabor Riesz sequences via the Ron-Shen duality principle.
For x 6 £2(Z2) and n 6 N we define the orthogonal projections Pn by
/р \	if max{|fc|,|/|} < n,
10 otherwise.
We now consider truncated versions of the operator H in (9.9),
Hn : L2(R)	f2(Z2), Hnf = PnHf,
which we identify with
Hn :L2(R) ^C(2n+1>\ Hnf = {(f,Ek/aTl/bg}}ww^n. (16.15) The matrix
НПЯ* = PnHH*Pn = {(Ev/arz7bS)Efc/aTz/bP)}|fc|,|Z|,|fc,|,|d<„, (16.16)
is a finite section of the infinite-dimensional matrix HH*. Motivated by (9.11) we let
7(n)	:= a6^(^H:)-1Pn{<5fc,o<51,o}Jfe)ieZ	(16.17)
= a(’H*(PnH*)-1{5/:io5i,o}|fc|,|/|<n for n G N.
We need a lemma before we present the main result in Theorem 16.5.2.
Lemma 16.5.1 Let К : £2(Z2) -> £2(Z2) be an invertible operator and cl sequence of positive bounded operators on £2(Z2) which converge strongly to K. If each operator Kn maps Pn^2(Z2) onto itself (thus, Kn restricted to this space is invertible) and there exists a constant C > 0 such that
CI < Kn on Pn£2(Z2), Vn e N, then
к-'РпХ -> k-'x, Vz e £2(z2).
Proof. Let x 6 £2(Z2). Then
•
We see immediately that |— PnK~lx\| -» 0 as n -» oo. Thus we have to show that also the second term converges to zero. Now, for any n G N, Theorem A.6.5 implies that K~} < ^1 on Pn£2(Z2), so
||(-Kn |p„Z2(Z’)) 'll < 77-
16.5 The finite section method
381
Thus
<	||(A\|p„£2(Z2))“,|| llKnPUf-'x-PnxIl
<	ill^nPn/V-'x-PnxIl
о
—> 0 as n —> oo.
□
We now prove that the functions in (16.17) indeed converge to 3~гд for n —> oo.
Theorem 16.5.2 Let g 6 L2(IR) and a,b > 0 be given, and assume that {EkbTiag}k,iez is a frame for L2(K). Then
zy(n) S~}g for n —> oo.
Proof. Letting A, В be frame bounds for {EkbTiag}kiigL, we know by Theorem 9.2.6 that {Ек/аТ1/Ьд}к}1(г% is a Riesz sequence with bounds abA, abB. In particular, for each finite scalar sequence {ckti},
2
abA \ck,i| <	ck,iEk/aTi/bg < abB \ck,i| •
|/c|,|/|<n	|/c|,|/|<n	|fe|,PI<n
In terms of the operator Hn this means that
\\HnH*\\ = \\H*\\2 > abA.
Since HnH* -> HH* strongly for n -> oo we can now apply Lemma 16.5.1 to conclude that
(НПЯ*)-1РП -> (ЯЯТ1 for n oo.
Now the definition of in (16.17) combined with (9.11) yields that
7(n) = abH*(HnH*n)-lPn{5kMktl&
->	= S~xg.
□
Strohmer also proves that the above method converges exponentially if g as well as g decay exponentially. That is, the assumptions imply that for some constants C", A' > 0,
||5-I9-7(n)|| <C'e~Xn
The proof uses Lemma 9.3.3 and is not constructive. It would be very useful to have knowledge of concrete values of C", A'.
Prior to Theorem 16.5.2 Strohmer proved similar results for approximation of the inverse frame operator associated to shift-invariant systems in £2(Z). We refer to [267] for details.
382	16. Approximation of the Inverse Frame Operator
16.6 Exercises
16.1 Prove that the near-Riesz basis in Example 16.1.3 is not a Rie: frame.
16.2 Prove that for an arbitrary frame {A}^i and n e N one can choose a set Jn such that (16.10) is satisfied.
17
Expansions in Banach Spaces
The material presented so far naturally splits in two parts: a functional analytic treatment of frames in general Hilbert spaces, and a more direct approach to structured frames like Gabor frames and wavelet frames. In this final chapter we make connections to abstract harmonic analysis and show how we can gain insight about frames via the theory for group representations. More precisely, we show how the orthogonality relations for square-integrable group representations lead to series expansions of the elements in the underlying Hilbert space; on a concrete level, this gives an alternative approach to Gabor systems and wavelet systems. Feichtinger and Grochenig proved that the group-theoretic setup even allows us to obtain series expansions in a large scale of Banach spaces, a result which leads Grochenig to define frames in Banach spaces. By removing some of the conditions we obtain p-frames, first studied separately by Aldroubi, Sun, and Tang.
This chapter is more advanced than the previous chapters. It is less detailed, and states open problems for future research.
17.1 Representations of locally compact groups
In this Section we let G denote a group with neutral element e. The composition of two elements x,y G G will be written x • у or simply xy. We will assume that G is equipped with a Hausdorff topology with respect to which e has a neighborhood whose closure is compact; such a group is said
384
17. Expansions in Banach Spaces
to be locally compact. We will always assume that Q can be covered by a countable union of compact sets, i.e., that Q is o-compact. We let (9(e) denote the family of neighborhoods of e, i.e., the sets V C Q containing e in the interior.
Every locally compact group Q can be equipped with a unique (up to scalar multiplication) positive measure p which is left-invariant in the sense that for all continuous functions F : Q -» C with compact support,
[ F(yx)dp(x) = f F(x)dp(x), Vy € Q.
Jq	jq
p is called the left Haar measure. The right Haar measure is defined similarly; if the right and left Haar measures coincide (after appropriate normalizations) we simply speak about the Haar measure, and Q is said to be unimodular.
The simplest example of a locally compact group is JRn equipped with the composition and the Euclidean topology. Another example is the torus
T = {z e C ] \z\ = 1};
here the composition is complex multiplication and the topology is inherited from C.
In physics and chemistry a large number of groups play important roles. Let us just mention the symmetry groups in chemistry, which are defined as the families of rotations in B3 which leaves a given molecule invariant. From the mathematical point of view they are subgroups of the group consisting of all invertible 3x3 matrices.
The elements in a group can be quite abstract objects, so it is desirable to transfer questions on a group into a more familiar setting. This is done by the concept of a group representation, which identifies (see the comment after the definition) the group elements with certain operators on a Hilbert space. In the example with symmetry groups one takes H = or H = C1 and identify the group elements with matrices.
Definition 17.1.1 Let Q be a locally compact group with left Haar measure p, and H a Hilbert space. A representation ofQ onH is a family of bounded invertible operators {7г(т)}хе^ on H for which
(i)	л(ху) = 7г(ж)7г(?/), \/x,y e Q.
(ii)	for all f G H the mapping x i-> Tv(x)f is continuous from Q into H.
We further say that
(iii)	tv is unitary if all the operators {tf(z)are unitary.
(iv)	tv is irreducible if the only closed subspaces of H which are invariant under all the operators {tv(x)}xEc; are {0} and H.
17.1 Representations of locally compact groups
385
(v)	A unitary irreducible representation тг is integrable if
I Jg	)
A square-integrable representation is defined similarly.
Condition (ii) (called strong continuity of 7r) is not always part of the definition. As said before, the idea behind a group representation is to identify elements in Q with operators. For this to hold, we also need the mapping x i-> 7г(я) to be injective; a representation with this property is said to be proper.
A representation 7Г is irreducible if and only if (Exercise 17.2)
span^a;)/}^ = H, V/ 6 ft \ {0}.
Assuming that 7Г is irreducible and fixing an arbitrary g 6 H\ {0}, we can thus approximate any f G H arbitrarily well by finite linear combinations of vectors 7r(z)g, x 6 Q. It is therefore very natural to ask if we can find g G H and a sequence {xk}^==1 in Q such that {тг^)#}^ is a frame. The answer turns out to be yes in a very general case if 7Г is an integrable representation. Before we present results in that direction we give some concrete examples of groups and their representations.
Example 17.1.2 The Heisenberg group is the set Q := Kx IR x T equipped with the product topology and the composition
(ai, 6i, tfi) • (<12, 62, ^2) — (&i 4- П2, 61 4- 62, U^2e27ribia2).
The Heisenberg group is not abelian, but it is unimodular: the Haar measure is the product measure of the three involved Lebesgue measures; see e.g. [172]. The special choice of the composition implies that we can define a representation of Q on L2(K) by
[тг(а, b, t)p](y) = te2*ib^g(y -a), gt L2(R),(a,6,t) € Q,y 6 R. (17.2)
This is the Schrodinger representation. To see that 7Г actually defines a representation, note that in terms of the operators Ea and Тъ from Section 2.5,
[тг(а, b, t)g](y) = te~2™b EbTag(y).
Using the commutator relations for the operators Ea and Тъ one can now prove that (i) of Definition 17.1.1 is satisfied (Exercise 17.3) ; (ii) follows by Lemma 2.5.1, which also shows that 7Г is unitary. It is less obvious that 7Г is irreducible and integrable, but this follows from our results about Gabor systems. To see this, let f G H \ {0} and assume that #±7r(a, 6, t)f for all
386	17. Expansions in Banach Spaces
(a,b, t) G Q. Then, by Proposition 8.1.2,
ZOO POO z*l
/	/ \{g, тг(а, b, t)f)\2dtdadb
-oo J — oo J0
ZOO POO
/ \{g,EaTbf}\2dadb
-OO J —oo
= ll/ll2 IK-
Therefore g = 0 and 7Г is irreducible. Note also that A = So, the Feichtinger algebra, which is dense in L2(K); thus 7Г is integrable.
Note that
[7г(па,тп6,1)р](2/) = е~2*гтпаЬЕтЪТпад(у), m,nEZ,
i.e., up to an irrelevant factor of absolute value 1, the Schrodinger representation sampled on the set {(na,mb, l)}m,nez and applied to g G L2(R) corresponds to the regular Gabor system {EmbTnag}
A technical detail: The torus component in the Heisenberg group will never play any practical role in this context. It is only introduced in order to obtain a group representation involving the operators EaTb, in fact, the operators defined by p(a,b) = EaTb do not form a representation of L2(K) on K2 (Exercise 17.2).	□
Example 17.1.3 The ax + b group is the set Q = К x К \ {0} equipped with the product topology and the composition
(5, a) • (ж, s) = (ax -I- b, as).
The left Haar measure is -^dadb and the right Haar measure is -^dadb, where dadb is the Lebesgue measure on tf2. In particular, the group is not unimodular. The name affine group is also used for the affine group; we refer to [151] for a more detailed discussion of its properties. One can define a unitary representation on L2 (R) by
= TbDaf(y) =	(b,a) &Q, ft L2(R), у £ R.
у/a a
Note that
[тг(Ька\ a^)g](y) = a~^2g(a~^y — kb), j,k G Z,
i.e., the wavelet systems appear by appropriate samplings of the representation. The representation satisfies the integrability condition (17.1), but is not irreducible. This is not a problem in practice: it is possible to extend the ax + b group to a larger group Q such that an appropriate extension of 7Г is a unitary irreducible representation satisfying the integrability condition. We shall not go into the technical details, but this is the reason that we still speak about this representation in the context of integrable
17.1 Representations of locally compact groups 387
representations (see e.g., the discussion on page 388). We refer to [142] for a description of the role played by irreducibility in the general case. □
Given a representation 7Г of Q on H we choose g G H and consider the transformation
v9 : H C&), Vg(J)(x) = (f, n(x)g).	(17.3)
Here C(G) denotes the set of continuous complex-valued functions on Q. With our convention for the inner product, Vg is a linear operator - but it depends conjugated linear on g. Note that if 7Г is the representation of the ax + b group considered in Example 17.1.3, then Vg(f) is the continuous wavelet transform of f with respect to g. For this reason the misleading word “wavelet transform” has also been associated to the transform in the general case. The correct terminology used in abstract harmonic analysis is that Vp(/) is a representation coefficient for the representation 7Г. Similarly, g has frequently been called the “mother wavelet”, while “analyzing atom11 or “generator” is more appropriate.
Our purpose is to show how an integrable group representation 7Г leads to expansions of the elements in the Hilbert space associated with 7Г, as well as in a class of related Banach spaces. The starting point is the orthogonality relations, first proved in [120] (see also [150]). They give an expression for the inner product between two representation coefficients in L2(Q):
Theorem 17.1.4 Let Q be a locally compact group with left Haar measure p, and assume that к is a square-integrable representation ofQ onH. Then there exists a unique positive self-adjoint operator U with domain T>(U), such that
(i) V9{g)eL\Q)^ge-D(U).
(ii) For all gi,gz 6 T>(U) and /ь/2 £ H,
[ (fi,Tr(x)gi){f2,Tr(x)g2)dn(x) = (Ug1,Ug2){f2,f1).	(17.4)
Jg
P(77) is dense in TL. If Q is unimodular, then V(U) = TL and U is a multiple of the identity on TL.
Theorem 17.1.4 immediately leads to continuous frames as discussed in Section 5.8:
Corollary 17.1.5 Let к be a square-integrable representation of Q on TL. Then, for all g 6 T>(U) \ {0}, {^(x)g}xEg is a tight continuous frame for TL (with respect to Q equipped with the left Haar measure). In particular, this holds for all g 6 TL\ {0} if Q is unimodular.
388
17. Expansions in Banach Spaces
Corollary 17.1.5 gives an abstract explanation of the differences we have observed between Gabor analysis and wavelet analysis: the WeyD Heisenberg group is unimodular and the Schrodinger representation is square-integrable, so all g 6 L2(K) \ {0} leads to continuous frames, in accordance with our direct proof in Corollary 8.1.4. On the other hand the ax 4- 6-group is not unimodular, so there might be g G L2(K) \ {0} which does not generate a continuous frame {т?(х)д}хед- This fact is expressed by the admissibility condition in Corollary 11.1.2.
It is worth noting that the two representation coefficients appearing in the orthogonality relations (17.4) might be with respect to different analyzing atoms #i,#2 in the transforms. In Proposition 11.1.1 we did not use this freedom: the same function was used for both of the appearing wavelet transforms. Additional freedom is actually obtained if we choose different functions. For example, we have only considered the integral (11.6) in the weak sense, but by putting different conditions on two functions , V>2 we can obtain pointwise convergence for a large class of functions, cf. [177]:
Proposition 17.1.6 Let ^1,^2 £ L1(K) and assume that
(I)	J-oo	l^2<7)l < oo;
(ii)	^2 is differentiable with G L1(JR);
(iii)	^(O) = ^2(0) = 0.
Assume that f 6 L2(K) is bounded and continuous in a point x € R. Then
17.2 Feichtinger-Grochenig theory
Feichtinger-Grochenig theory was presented in a series of papers appearing around 1990. The purpose of the papers was to obtain series expansions in a large class of Banach spaces based on the theory for integrable group representations. We will give a short introduction to the theory and refer to [128], [129] and [154] for more details and further results.
Let Q denote a locally compact group, equipped with a left Haar measure /z. Define the translation operator Tx, r E S, acting on functions f : Q C, by
(2W)(v) = у e Q-
In the generality discussed here, Tx is called the left regular representation', in the special case Q = К it equals our translation operator in Section 2.5.
17.2 Feichtinger-Grochenig theory
389
For functions F,G E L1^), the convolution F * G : Q -» C is defined by F*G(i/) := [ F^G^y^x)
F(x)TxG(y)dp{x), yeG.
The assumption F, G E L1^) implies that F * G is well defined and belongs to L1^). However, the convolution is well defined under many other conditions on F,G, and we will use the convolution symbol for any pair of functions F, G for which F * G is a well-defined function.
If the function F does not oscillate too much, the convolution F *G can be considered as the limit of a sequence of linear combinations of translates of the function G, with weights determined by F.
Lemma 17.2.1 below relates convolution and the orthogonality relations. It gives a formulation of the orthogonality relations which is very convenient for our purpose, and which is the starting point for Feichtinger-Grochenig theory.
Lemma 17.2.1 Let tv be a square-integrable representation of Q on FL. Then the following holds:
(i)	= Tvvg(f), VfeKyeg.
(ii)	The operator U introduced in Theorem 17.1.4 is injective.
(iii)	Choosing g E F(F) such that ||Fg|| = 1, we have
W) = b(W),v/eW,	(17.5)
and the orthogonal projection of L2(Q) onto Ttyg is
FhF*V9(9( F E L2(£).	(17.6)
Proof, (i) follows by a direct calculation. To prove (ii) we let g E P(F), and assume that Ug = 0. Via the orthogonality relations we see that for //0,
o = ||77p||2 = i|4j|2 [ 11J 11 J Q
by continuity of the representation coefficients we conclude that
(/, тг(х)д) = 0 for all x E Q.
Since this holds for all f 0, we have 7r(z)g = 0 for all x E £7, and therefore 0 = 0.
390	17. Expansions in Banach Spaces
For the proof of (iii) we first show that F * Vg(g) is well defined for any F 6 L2(Q). Note that by (i) and the left invariance of the Haar measure, [	= [ [V^y^x^d^x)
Jg	Jg
|y<,(p)(a;)|2d^(x)
< oo,
i.e., Vg(xr(y)g) e L2(£). Since
F(x')Vg(g)(x~1y') = F(x)(g,Tr(x~vy)g') = F(x)(it(y)g,Tt(x)g) = F(x)Vg(n(y)g)(x),	(17.7)
it follows that
x H- F(x')Vg(g)(x^ly)
is integrable for у E Q, i.e., that F * Vg(g) is well defined.
As a consequence of (ii), an arbitrary g 6	\ {0} can be normalized
such that ||C7p|| = 1. Doing so, and applying the orthogonality relations with gi = g2 = g, fi = тг(y)g and /2 = / for an arbitrary / G %,
v9(f)(y) = (ir(y)g), тг(х)д) {f, ir(x)g)dp,(x)
(g,7r{x~1y)g){f,7r(x)g)dn(x') = Vg(f)*Vg(g)(y).
This proves in particular that F i-> F * Vg(g) is the identity on Fyg. For the second part of (iii) we only need to prove that F * Vg(g) = 0 for all F belonging to the orthogonal complement of Fyg in L2(Q). But for these F, (17.7) shows that
- [ F{x)Vg(g)(x-iy)dfx(x) Jg
= {^Уд(тг(у)д)) = 0.
□
The result in (i) is expressed by saying that Vg is an intertwining operator for the representations 7г(ж) and Tx.
Formula 17.5 gives an integral representation of all functions Vp(/), W)(2/) = Vg(J) * Vg(g)(y) = f Vg(f)(x)TxVg(g)(y)dli(x), feKy^.
17.2 Feichtinger-Grochenig theory 391
With our interpretation of the convolution on page 389 it is natural to search for a representation of F = Vg(f) via an infinite linear combination of translates of Vg(g)’, formulated in short, to search for expansions
oo
F = ^Ck{F)TXkVg{g), FelZVs.	(17.8)
fc=l
The question is how to choose the points {xk	in Q and the coefficient
functionals Cfc. We will base the choice of	on our knowledge from
series expansions via a Gabor frame.
Recall from Theorem 8.3.1 that in order for {ЕтьТпад}minez to be a Gabor frame, it is necessary that ab < 1. In terms of the Schrodinger representation of the Weyl-Heisenberg group, {£Jm{,Tna^}m)nez corresponds (up to a constant) to {тг(па,т5,l)^}m,nez, so the condition ab < 1 means that the points {(na,rnb)g}minEz have to be “sufficiently dense” in K2. On the other hand, for an irregular Gabor family {^(x^yk, l)p}fc^=i to be a frame the points {(я&,Ук)}%^ are not allowed to be too dense (Exercise 17.4). Motivated by these considerations we introduce some definitions related to general locally compact groups.
Definition 17.2.2 Let	be a sequence in Q.
(i) Let V E 0(e) be relatively compact. If UxkV = Q, then	is
said to be V-dense.
(ii) If there exists a relatively compact neighborhood V 6 0(e) such that XkV П XjV = 0 for к j, then	is said to be separated.
{zk}kTi is relatively separated if it is a finite union of separated sets.
For the definition of appropriate coefficient functionals in (17.8) we need some special types of partition of unity:
Definition 17.2.3 Let V 6 O(e) be compact. A family Ф =	of
continuous functions on Q is a partition of unity with size V if
(i) 0 < ipk(x) < 1 and V’k(z) = 1, Vz € Q.
(ii) There exists a relatively separated and V-dense set	in Q for
which supp ifk Q XkV, Vk 6 N.
It is important to notice that such partitions of unity can be constructed for arbitrarily small neighborhoods V 6 O(e) (see [125]).
A special case of the main result in [128] says:
Theorem 17.2.4 Let Q be a unimodular locally compact group and 7Г an integrable representation of Q on H. Given g E A \ {0}, there exists a
392
17. Expansions in Banach Spaces
neighborhood V E 0(e) with the following property: for every V-dense and relatively separated family	there exists a bounded operator
A:77-+£2(N), A/ = {Afe(/)}^=1,
such that
oo
f = ^MfM^g, Vfe H.	(17.9)
Proof. We will not give a full proof but only sketch the main points. The basic idea is to approximate the convolution operator F i-> F * Vg(g) by operators of the type
oo
Сф : KVg -4 KVg, C*(F) = ^F,^k)TXkVg(g),
where Ф =	is a partition of unity of size V. One can prove that
if V is chosen small enough and	denotes a set of points in Q as in
Definition 17.2.3, then Cy(F) is well defined and for some constant C < 1,
||F*V}(S)-C4(F)|| <C||F||, VFeTJv,.
Since F * Vg(g) is the identity on the Banach space Fvg, this implies that Оф is invertible on Куд; thus each F € Fyg has a representation
oo
F = C*C^(F) =
That is, for f e 7/,
oo
Applying the intertwining property in Lemma 17.2.l(i), we obtain a representation of f G H'.
oo
/ = Vw = Е(с;Ш)).ж1г‘Л(9)
fc=l
oo
k=l
The proof that f —>	*фк)}к^1 is bounded from H into ^2(N)
can be found in [128].	□
Corollary 17.2.5 The conditions in Theorem 17.2.4 imply that
is a frame for TL.
17.2 Feichtinger-Grochenig theory
393
Proof. We only verify the lower frame condition and refer to [129] for the proof that {7t(xk)g}kLi is a Bessel sequence. Let f G TL. Then, putting the expression for f from Theorem 17.2.4 into the first entry of (/, f), we obtain that
Г°°	Г
ll/ll4 =	ЕЛ*(/)(ТГ(Х*)Р,/)
.k=l
oo	oo
<	E|Ak(/)|2 ЕКтгОШ/)!2
oo
<	или2 н/п2 Eiw^jp.ni2,
fc=l
from which the result follows.
□
When we apply Theorem 17.2.4 to the Heisenberg group and the Schrodinger representation we obtain a result about irregular Gabor frames. For the proof we only need to recall from Example 17.1.2 that in this case A equals the Feichtinger algebra 50.
Corollary 17.2.6 Let g G So\ {0}. Then there exists an open set V С B2 such that {ЕхкТр,кд}<^=1 is a frame for L2(IR) for every separated sequence {(jak, Afc)}£Li in IR2 for which
oo
U[(m, Aft) + v] - R2. /c=l
This short description is far from giving full justice to the work by Feichtinger and Grochenig; we will now mention a few central points where the theory is more general than described here.
From the sketch of the proof of Theorem 17.2.4 it is not clear why Q needs to be unimodular, and it is in fact an unnecessary assumption. However, without this assumption we need to be slightly more restrictive with the choice of g 6 TL The class of usable g G TL is still dense in TL, but its definition is slightly more involved; see [128].
Of even more importance is the fact that Feichtinger-Grochenig theory extends to series expansions in a scale of Banach spaces. The key point in [128] is the observation that the convolution identity (17.5) can be extended to hold for a large class of distributions /; here the notation
v^(/) = </,7г(х)р>
has to be reinterpreted as the action of the distribution f on тг(х)д. To a large class of Banach function spaces Y (including weighted //-spaces) one can associate a sequence space Yd and a Banach space СоУ such that Theorem 17.2.4 holds with TL replaced by СоУ and Z2(Z) replaced by У^. The
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396
17. Expansions in Banach Spaces
Already in Chapter 3 we mentioned that there exist separable Banach spaces having no basis. From this point of view one could say that the concept of Banach frames is very satisfying because they always exist. However, one could also be suspicious and ask if the pure existence of Banach frames is interesting. In order for a Banach frame to be practically useful, it has to be defined with respect to a convenient and easily identifiable sequence space Xd\ this is not the case with the Banach frame constructed in the above proof. One should rather ask for the existence of Banach frames with respect to a nice class of sequence spaces, which would make Banach frames share more of the properties we know from frames in Hilbert spaces. This point of view is supported by a result by Stoeva, to be published in [56]: it says that every total sequence in X* is a Banach frame for X with respect to some sequence space Xd- This gives a strong argument for restricting the class of Banach frames to consider. For example, let be an orthonormal basis for a Hilbert space %, and consider the family {ek 4- ek+i}^-! in Example 5.4.6: then
•	{ek 4- efc+i}^=1 is a Banach frame for %;
•	{ek 4- ek+i Ijg-j is not a frame for H.
The existence of such examples shows that the definition of Banach frames does not necessarily match the definition of frames. The question of the “right definition” of a Banach frame is currently investigated. Feichtinger and Grochenig have recently advocated to use some special frames in Hilbert spaces as the starting point: in fact, there exist frames which are at the same time frames for a scale of Banach spaces. We will see a concrete example in the discussion on page 398. A general framework was recently developed by Grochenig [156], who introduced localized frames in Hilbert spaces. To such a frame one can associate a class of Banach spaces, and all the central frame objects carry over from the Hilbert space to these spaces. In particular, the frame operator extends to a bounded bijection on each space, which leads to frame decompositions exactly as in Theorem 5.1.6. In concrete cases the Banach spaces correspond to e.g., modulation spaces, Besov spaces, or shift-invariant spaces (see again page 398).
Suppose that the conditions (i) and (ii) in Definition 17.3.2 are satisfied. Then the operator
U . X -4 Xd, Uf := {9k(fW=i
is injective, and thus has an inverse
U-1 : C Xd X,	= f, П X-
The only condition that is missing in order for ({pfe}^, t/-1) to be a Banach frame is that the operator U~1 can be extended to an operator on Xd- If TZu is complemented in X/, the operator U~1 can be extended by zero on a complement; however, no easily verifiable condition for a subspace
17.4 p-frames
397
to be complemented exists in general, even in the case where Xd is an £p-space. Thus, it is in general not very fruitful to try to obtain Banach frames this way.
17.4 p-frames
The first part of the definition of a Banach frame was considered separately by Aldroubi, Sun and Tang [4] in the case where Xd is an £p-space:
Definition 17.4.1 Let p G]l,oo[ be given. A family	С X* is a
p-frame for X if there exist constants А, В > 0 such that
/ oo	\ VP
А||Ж< ЕЫЛ1Р <B Ilfllx, \/f ex. (17.10) \k=l	/
is a p-Bessel sequence if at least the upper p-frame condition is satisfied.
In [4] p-frames are used to obtain series expansions in shift-invariant subspaces of LP(R). Let
W=	supV|Tfc/(z)| <ooL
I	xeRkez	)
If ф e W, then	converges in LP(JR) for all {ck}kez € ^(Z),
and we can consider the space
Sp~ j£>fcTW| {ck}k& € ^(Z) I. L к GZ	J
Note that the case p — 2 appeared in Section 7.5. The space Sp is said to be shift-invariant because
f e Sp => Tkf e Sp, Vk e z.
The main result in [4] shows a strong similarity between p-frames and frames:
Theorem 17.4.2 Ье1ф 6 W andp G]l,oo[. Then the following statements are equivalent:
(i)	Sp is closed in LP(JR).
(ii)	{Ткф}ке2 is a p-frame for Sp.
398
17. Expansions in Banach Spaces
(iii)	There exists a function ф E W such that each f 6 Sp has unconditionally convergent expansions
f = ^,Ткф}Ткф = ^,Ткф)Ткф. kez	kez
The original article is more general than stated above and it applies to a space Sp generated by a finite collection of functions rather than the single function ф. The cases p = l,oo are covered by requiring that ф belongs to the Wiener space W, which is a stronger condition than membership of W. If ф e W it is also proved that the conditions in Theorem 17.4.2 are independent of the choice of p; in particular, if {Т^ф}кег is a p0-frame for one value of p0, it is a p-frame for all p 6 [l,oo]. This proves for instance that one automatically obtains series expansions like (iii) in a large scale of Banach spaces simultaneously if one can prove that {Ткф}кё£ is a frame sequence in L2(]R). Thus, frame theory in Hilbert spaces provides a convenient way to obtain expansions in some Banach spaces.
The definition of Riesz bases can also be extended to Banach spaces. The definition will usually be applied in the dual of X, so in order to avoid confusion we state it in a Banach space Y. Furthermore, as a standard convention we let q denote the conjugated exponent of p, i.e.,
Definition 17.4.3 Let q G]l,oo[ be given, and let Y be a Banach space. A family	C Y is a q-Riesz basis for Y if span {p/J^ = Y and
there exist constants А, В > 0 such that for all finite scalar sequences {dk}, .4 (E 1*1')'''s I IE ** | |r £ B (E i*i') <17-ч> Note that completeness is part of our definition of a p-Riesz basis (in contrast to the definition in [4]). Standard arguments show that the assumptions in Definition 17.4.3 imply that dkQk converges unconditionally for all {dk} G £Q and that (17.11) holds also for these sequences (Exercise 17.7).
In the rest of this section we discuss results by Christensen and Stoeva [84]. First, note that if X can be equipped with ap-frame, then X is isomorphic to a closed subspace of £p and therefore reflexive. A characterization of the p-frame property is given by
Theorem 17.4.4 Let X be a reflexive Banach space and	C
Then	is a p-frame for X if and only if
oo fc—1
is a well-defined mapping of £g onto X*.
17.4 p-frames 399
Note that Theorem 17.4.4 does not mean that the p-frame property is enough to obtain frame-like expansions in X*: the result only says that if {gk}™^ is ap-frame, then each g e X* has a representation g = dkgk for some {dk}^-! € but nothing guarantees that the coefficients {dk}^-^ can be chosen as continuous linear functionals on X*.
Theorem 17.4.4 sheds some light on the reason for adding the condition (iii) to the definition of a Banach frame, see Definition 17.3.2: it shows that in the special case of Xd — £p the norm-equivalence in (ii) alone is equivalent to some kind of “expansion property” in X*, which is clearly different from obtaining expansions in X.
Corollary 17.4.5 Let {дк}^^ С X* be a q-Riesz basis for X* with q-Riesz basis bounds A,B. Then {gk}kLi is a p-frame for X with p-frame bounds A and B.
For Q-Riesz bases, the type of expansions we are interested in exist without further assumptions, in X as well as in X*:
Theorem 17.4.6 If {д^}^! С X* is a q-Riesz basis for X* with bounds A,B, there exists a unique p-Riesz basis {fk}^-i С X for X for which
oo
/ =	V/GJf,	(17.12)
k=l
oo
9 =	(17.13)
k=l
{fk}kLi has the bounds j?,
In [56] it is proved that for some Banach spaces X and p 2, there exist p-frames	for X, for which no family {fk}^=i C %. satisfies that
oo
/ = Е^(/)Л, V/6 x.	(17.14)
k-\
Since a Q-Riesz basis for X* is a special case of a p-frame for X, Theorem 17.4.6 suggests the following question: given a p-frame	С X* for
X, under what conditions can we find a Q-frame {fk}kLi С X for X* such that (17.14) is satisfied? A theoretical answer is contained in the following theorem.
Theorem 17.4.7 Suppose that {gk С X* is a p-frame for X. Then the following are equivalent:
(i)	Ru is complemented inlp.
(ii)	The operator U^1 : IZu -» X can be extended to a bounded linear operator V : Ip -> X.
400
17. Expansions in Banach Spaces
(iii)	There exists a q-Bessel sequence {fk}kLi С X for X* such that oo
J =	V/ ex.
k=l
(iv)	There exists a q-Bessel sequence {fk}kLi С X for X* such that
g = Xg(fk)gk, ^ex‘.
k=l
(v)	{9k }/Xi is a Banach frame for X with respect to £p.
If (one of) the conditions are satisfied, the sequence	in (iii) is a
q-frame.
17.5 Gabor systems and wavelets in 77(R) and related spaces
The Lp-spaces are for p ф 2 not the right spaces to search for unconditionally convergent Gabor expansions. Feichtinger-Grochenig theory leads to unconditionally convergent expansions in coorbit spaces, but in [132] it is proved that LP(JR) is not a coorbit space under the Schrodinger representation for p 6 [l,oo[\{2}. The “right spaces” in connection with Gabor analysis are the modulation spaces as described on page 394, and explained in detail in [153].
If one is satisfied with conditionally convergent Gabor expansions one can obtain convergence for functions in LP(R) by requiring that the generator for the Gabor frame as well as the generator for the canonical dual frame belong to the Wiener space, cf. [157]:
Theorem 17.5.1 Letp G]l,oo[, g G W anda,b > 0 be given. Assume that {EmbTnag}mine% is a frame and that S-1g G W. Then, for an arbitrary feLp№),
^2 22 ЕтъТпаЗ~1д)ЕтЬТпад -4 f in LP(R) as N -4 oo.
|m|<N |n|<W
For wavelet systems only little work has been done on overcomplete systems in LP(JR). It is well known that a large class of wavelet orthonormal bases in L2(K) are unconditional bases for LP(K) for all p G]l,oo[ (most wavelet books contain versions of this statement), but the analogues for overcomplete systems have apparently not been studied yet. Some steps have, though, been taken: for example, Chui and Shi find in [94] sufficient conditions for the frame operator for a Bessel sequence {2^2^(2Jx — k)}3}kEi in L2(JR) to extend to a bounded operator on Lp« pE]l,oo[. ’
17.6 Exercises
401
17.6 Exercises
17.1	Prove that every abelian locally compact group is unimodular.
17.2	Prove that a representation 7Г is irreducible if and only if spanMz)/}ze£ = 7/, V/ EH\ {0}.
17.3	Prove that 7Г defined by (17.2) satisfies condition (i) in Definition 17.1.1. Prove also that the operators p(a, b) = ЕаТь do not form a representation of (Ж2, Ч-) on L2(R).
17.4	Prove that if {(#*?, 2/fc)}^ C IR2 has an accumulation point, then the samples of the Schrodinger representation {7г(ж^, yk, 1)#}^ can not be a frame for any g 6 L2(R).
17.5	Prove that an integrable representation is also square-integrable.
17.6	Let 7Г be an integrable representation. Show that the set A defined in (17.1) is dense in H.
17.7	Show that the assumptions in Definition 17.4.3 imply that dkQk converges unconditionally for all {dk} G £g and that (17.11) holds for {dk} € £q.
Appendix A
A.l Normed vector spaces and inner product
spaces
Throughout this Section we let V be a complex vector space over C
A norm on V is a function || • || : V -» [0, oo[ satisfying
||z|| = 0 О x = 0,
||az|| = |a| ||z||, \fx 6 V, a 6 C,
ll^-ЬЗ/Ц < INI + Из/ll, ^x,y 6 V.
If V is equipped with a norm, we say that V is a normed vector space. The opposite triangle inequality is satisfied in any normed vector space:
Ik -vll > I Ikll - IMI I, x,y e v.	(A.i)
An inner product on V is a function (•, •) : V x V -» C for which
(i)	(x,y) = (y,x), \fx,y e V.
(ii)	(ax 4- Py.z} = a(x,z) 4- /3{y,z), Vx,y,z eV,a,/3 e C.
(iii)	(x, x) > 0, V# 6 V, and (x, x,) = 0 <=> x = 0.
Note our convention (ii) of taking the inner product linear in the first entry. It implies that the inner product is conjugated linear in the second entry. Frequently the opposite convention is used in the literature.
404
Appendix A
A vector space with inner product can be equipped with a norm, namely ||z|| =	x e V.
Furthermore, Cauchy-Schwarz’ inequality holds:
l(ac,v)l < Ikll lll/ll, x,yeV.
Two elements x,y e V are orthogonal if (x,y) = 0; and the orthogonal complement of a subspace U of V is
UL = {x eV : (x,y) = 0, Vi/ G U}.
The above definitions and results are valid whether V is finitedimensional or infinite-dimensional. Also note that norms and inner products are defined in a similar way on real vector spaces (just replace the scalars C by the real scalars IR). Throughout the book we assume all vector spaces to be complex, except in a few illustrating examples taking place in
A.2 Linear algebra
Let V, W be finite-dimensional vector spaces, equipped with inner products (•, -)v, respectively (•, -)vv (when it is clear from the context in which space the inner product is taken we will skip the subscript). Assume that
dimV = n, dimW7 = m.
Assume that T : V —> W is a linear map, and that we have fixed an orthonormal basis {efcjJLj in V and an orthonormal basis	in W.
The matrix of T with respect to the chosen bases is the m x n matrix, where the A-th column consists of the coordinates of the image under T of the A;-th basis vector in V, in terms of the given basis in W. The j/c-th entry in the matrix representation is (Tek,ej)-
The matrix representation gives a convenient way to find the action of the linear map T on a given v G V: by writing v =	the result
of multiplying the matrix representation of T with {c^ }£_2 is the sequence of coordinates representing Tv in the basis for W. We will always identify the linear map and its matrix representation.
Given a linear operator T : V -» W, the adjoint operator T* : W -» V is characterized by
{Tx,y} = (х,т*у), x ev,y ew.
In matrix language, T* is represented by the hermitian transpose of the matrix for T, i.e., the matrix we obtain by complex conjugation and transposing.
The kernel for T is
MT = {xeV \ Tx = 0},
3 Integration
405
and the range is
тгт = {Tx |z e V}.
The vector spaces Л/т and TZt* are subspaces of V, and
Л/т — Ry*",
in particular, the linear map T induces orthogonal decompositions of V and (via T*) W given by
V =	(A.2)
w = л<т-Ф7гт.	(А.з)
In case T = T* (this can only happen when V = W), we say that T is self-adjoint. The finite-dimensional version of the Spectral Theorem says that a self-adjoint operator has enough eigenvectors to span the entire space:
Theorem A.2.1 If T : V -» V is self-adjoint, then all eigenvalues are real, and V has an orthonormal basis consisting of eigenvectors for T.
A.3 Integration
Here we state some basic facts from the theory of integration. The proofs and further results can be found in any standard book on the subject, e.g., [250].
Let X be a set and Л4 a сг-algebra of subsets of X, on which there is defined a positive measure p. An example is the real numbers IR with the Borel subsets as сг-algebra, and the Lebesgue measure. Another example is the natural numbers N equipped with the сг-algebra consisting of all subsets, and the counting measure.
A null-set is a measurable set with measure zero. A condition holds almost everywhere (abbreviated a.e.) if it holds except on a null set.
We begin with Fatou’s Lemma:
Lemma A.3.1 Let fn : X —> [0,oo], n 6 N be a sequence of measurable functions. Then
/ lim inf fndp < lim inf / fndp.
j X n—ioo	71—400 JX
Lebesgue’s Dominated Convergence Theorem is the main tool to interchange sums and integrals:
Theorem A.3.2 Suppose that fn : X -> С, n E N is a sequence of measurable functions, that fn(x) —> f(x) pointwise, and that there exists
406 Appendix A
a positive, measurable function g such that |/n| < g for all n E N and fx gdp < oo. Then
Lemma A.3.3 Let p be a positive measure on a a-algebra Л4. Assume that	С M and
Л D A2 Э • • • Э An Э ....
If /x(Ai) < oo, then
(oo \
A 4 = lim pfAn).
• 1 I	n—>oo
n=l /
A.4 Some special normed vector spaces.
1)	Given a family of Hilbert spaces	their direct sum ie denoted
by
(A-4)
by definition, H consists of all sequences g — (gi, g2, .. .) for which gn E Hn for all n E N, and ||gn||2 < oo. H is a Hilbert space with respect to the inner product
(Ла) =	f,g g H;
n=l
the associated norm is
oo
ik = E1Ы12-
n=l
2)	Given a parameter s > 0 we define the Sobolev space
HS(W = (/ : R -4 С I Г |/(7)|2(1 + |7|2)^7 < oo) .	(A.5)
I	J — oo	J
Hs(№) is a Banach space with respect to the natural norm, aoo	\ 1/2
J/(7)|2(l + l7|2r<A)	
A.5 Operators on Banach spaces 407
A.5 Operators on Banach spaces
Let X, Y denote Banach spaces. An operator is a linear map U : X -> У, and U is bounded or continuous if there exists a constant К > 0 such that
||СЛг||у <К|И|х, VrrGX.	(A.6)
Usually it will be clear from the context which norm we use, so we will write || • || for both || • ||x and || • ||y. The norm of the operator U, \\U\\, is the smallest constant К that can be used in (A.6). Alternatively,
||(7|| = sup {||[7x|| : x e Х,|И| = 1} .
If U\ and U2 are operators for which the range of U2 is contained in the domain of Ui, we can consider the composed operator U1U2; if Ui and U2 are bounded, then also U1U2 is bounded, and
imii < 11^11 ii^ii-	(a.7)
Now consider a sequence of operators Un : X -4 У, n € N which converges to a mapping U : X —> T pointwise, i.e.,
Unx -4 Ux, as n -4 00, Vx G X.
We say that Un converges to U in the strong operator topology. The Banach-Steinhaus Theorem, also known as the uniform boundedness principle, states the following:
Theorem A.5.1 Let Un : X -4 У, n G N, be a sequence of bounded operators, which converges pointwise to a mapping U : X -4 У. Then U is linear and bounded. Furthermore, the sequence of norms ||Un|| is bounded, and\\U\\ < liminf ||Un|(.
An operator U : X -4 Y is invertible if U is surjective and injective. For a bounded, invertible operator, the inverse operator is bounded:
Theorem A.5.2 A bounded bijective operator between Banach spaces has a bounded inverse.
In case X = У, it makes sense to speak about the identity operator I on X. The Neumann Theorem states that an operator U : X -4 X is invertible if it is close enough to the identity operator:
Theorem A.5.3 If U : X -4 X is bounded and \\I — U\\ < 1, then U is invertible, and U~r =	— U)k. Furthermore, ||U-1|| < -j—., T-f,..
A special role is played by the continuous linear operators U : X -4 C; they are called functionals, and the collection of all functionals is the dual X* of X.
408 Appendix A
A.6	Operators on Hilbert spaces
Let U be a bounded operator from the Hilbert space (/C, (•, into the Hilbert space (TL, (♦, we will usually write (•, •) for both inner products. The adjoint operator is the unique operator U* : TL -» /С satisfying
(x, Uy)H = (U*x,y}lc, \/x 6 TL,y G /С.
We collect some relationships between U and U*; the proofs can be found in e.g., [251].
Lemma A.6.1 Let U : KL -+TL be a bounded operator. Then the following holds:
(i)	I|t/|| = 11^*11 and = HOI2.
(ii)	1Zu is closed in TL if and only if Ku* is closed in KL.
(iii)	U is surjective if and only if there exists a constant C > 0 such that
lltn/ll > c IMI,
An operator U : /С -» TL is compact if V := {Ux : ||z|| < 1} is compact, i.e., if every sequence from V has a convergent subsequence. A compact operator is bounded. Among the compact operators we find all operators having finite rank, i.e., a finite-dimensional range. We collect some of the most important properties of compact operators; again, the proofs are in [251]:
Lemma A.6.2 Let U : /С -» TL be a compact operator. Then
(i)	The composition of U and a bounded operator (from left or right) is a compact operator.
(ii)	The adjoint operator U* is compact.
(iii)	If KL = TL and A 0, then U — XI has closed range; here I denotes the identity operator on TL.
The linear functionals play a special role. They are characterized in Riesz’ Representation Theorem:
Theorem A.6.3 Let f : TL -» C be a continuous linear mapping. Then there exists a unique у 6 TL such that f(x) = (x,y).
Corollary A.6.4 The dual of a Hilbert space TL can be identified with TL.
In the rest of this section we consider the case KL = TL. A bounded operator U :TL -+TL is unitary if UU* = U*U = I. If U is unitary, then
(Ux,Uy) = (x,y), Vx,y e TL.
A.6 Operators on Hilbert spaces
409
A bounded operator U : 7Y —> 7/ is self-adjoint if U = U*. When U is self-adjoint,
||q|= sup	(A.8)
l|x||=l
For a self-adjoint operator [7, the inner product (Ux,x) is real for all x E H. One can introduce a partial order on the self-adjoint operators by
Ui < U2 О (U\x,x} < (U2x,x), Vz e H.
Using this order one can work with self-adjoint operators almost as with real numbers, except that the operators might not commute. Concerning multiplication we state a result which is proved in e.g., [174]:
Theorem A.6.5 Let U\,U2,U3 be self-adjoint operators. If Ui < U2, U3 > 0, and U3 commutes with Ui and U2, then U1U3 < U2U3.
An important class of self-adjoint operators consists of the orthogonal projections. Given a closed subspace V of H, the orthogonal projection of H onto V is the operator P :H -+H for which
Px = x, x E V, Px = 0, x E V1.
If	is an orthonormal basis for V, the operator P is given explicitly
by
00
Px =
k=l
In case H is a complex Hilbert space and U is a bounded operator on a direct calculation gives that
4{Ux,y) = {U(x + y),x + y) — {U(x — y),x — y}
+i{U(x 4- iy), x 4- iy) — i{U(x — iy),x — iy).	(A.9)
In particular we can recover the inner product in H from the norm by
4(z,3/> = ||z4-?/||2- ||z- 2/|[2 4- г ||z 4-12/Ц2 -i ||z-z?/||2, a result which is known as the polarization identity.
Lemma A.6.6 Let U : H -» H be a bounded operator, and assume that (Ux,x) = 0 for all x EH. Then the following holds :
(i) IfH is a complex Hilbert space, then U = 0.
(ii) IfH is a real Hilbert space and U is self-adjoint, then U = 0.
Proof. If H is complex, we can use (A.9); thus, if (Ux,x) = 0 for all x E H, then (Ux, у) = 0 for all x, у E H, and therefore U = 0.
410 Appendix A
In case H is a real Hilbert space we must use a different approach. Let {efc}£Li be an orthonormal basis for H. Then, for arbitrary j, к e N,
о = (U(ek + ej),efc + e3) = {Uek,ej) + (Uej,ek} = 2{Ue3,ek)-,
therefore U = 0.	□
Note that without the assumption U = U*, the second part of the lemma would fail; to see that, let U be a rotation of 90° in Ж'2.
A bounded operator U : H -» H is positive if (Ux, x) >0, Vx G H. On a complex Hilbert space, every bounded positive operator is self-adjoint. For a positive operator U we will often use the following result about the existence of a square root, i.e., a bounded operator W such that W2 = U:
Lemma A.6.7 Every bounded and positive operator U : TL —> TL has a unique bounded and positive square root W. If U is self-adjoint, then W is self-adjoint. IfU is invertible, then W is also invertible. W can be expressed as a limit (in the strong operator topology) of a sequence of polynomials in U, and commutes with U.
Frequently, the study of an operator is easier if it can be represented as a sum or product of “simple” operators. We mention a few examples of such representations:
Lemma A.6.8 Let TL be a Hilbert space. Then:
(i) Every bounded and invertible operator U : TL —> TL has a unique representation U — WP, where W is unitary and P is positive.
(ii) Assume that TL is complex. Then every positive operator P on TL with ||P|| < 1 can be written as an average of unitary operators, namely
P = 1(IV + iv*) with W = P + iy/l - P2.
The representation U = WP in (i) is called a polar decomposition. The representation in (ii) is probably less known, but it is proved by direct verification. That W = P + iy/I — P2 is unitary follows by calculating WW* and WW* using that the square root of I - P2 can be considered as a limit of polynomials in I — P2 and therefore commutes with P. Note that (ii) applies if P is an orthogonal projection.
A.7 The pseudo-inverse
It is often desirable to find some kind of “inverse” for an operator which is not invertible in the strict sense. The following lemma gives a condition that ensures the existence of a “right-inverse”:
A. 7 The pseudo-inverse	411
Lemma A.7.1 Let be Hilbert spaces, and suppose that U : К, -» H is a bounded operator with closed range Hu . Then there exists a bounded operator W : H —> K, for which
UU'f = f,VfE7Zu.	(A.10)
Proof. Consider the restriction of U to an operator on the orthogonal complement of the kernel of U, i.e., let
U:=U^
Clearly U is linear and bounded. U is also injective: if Ux = 0, it follows that x 6 Uy AA/y = {0}. We now prove that the range of U equals the range of U. Given у 6 Hu, there exists x G K, such that Ux = y. By writing x = Xi + x2, where Xi 6 A/^, x2 G А4/, we obtain that
Uxi = Uxi = U(xi 4- x2) = Ux = y.
It follows from Theorem A.5.2 that U has a bounded inverse
t/-1 : 1ZV -> A#.
Extending U~x by zero on the orthogonal complement of Hu we obtain a bounded operator U^ : H -» K, for which UU^ f = f for all f 6 Hu .	□
The operator U^ constructed in the proof of Lemma A.7.1 is called the pseudo-inverse of U. In the literature one will often see the pseudo-inverse of an operator U with closed range defined as the unique operator satisfying
ЛГи<=ПЬ, Kut andWt/ = fj ETZu-, (A.ll)
this definition is equivalent to the above construction (Exercise 5.13). We collect some properties of U^ and its relationship to U.
Lemma A.7.2 Let U : K, —> Tl be a bounded operator with closed range. Then
(i)	The orthogonal projection of H onto Hu is given by UW.
(ii)	The orthogonal projection of JC onto Hut is given by UW.
(iii)	U* has closed range, and (£/*)t = (t/^)*.
(iv)	On Hu, the operator W is given explicitly by
U] = U^UU*)-1-	(A.12)
Proof. All statements follow from the characterization of U^ in (A.ll). For example, it shows that
UU^ = I on Hu and that UU^ = 0 on A/^t = itfj;
this gives (i) by the definition of an orthogonal projection. The proof of (ii) is similar. That Hu* is closed was stated already in Lemma A.6.1; thus
412
Appendix A
(C/*)t is well defined. That (U*^ equals (lP)* follows by verifying that (U^y* satisfies (A. 11) with U replaced by U*, and using the uniqueness. Finally, UU* is invertible as an operator on TZu, and the operator given by
U^UU^-1 on Пи, and 0 on
satisfies the conditions (A.ll) characterizing W.	□
The pseudo-inverse gives the solution to an important optimization problem:
Theorem A.7.3 Let U : /С -» H be a bounded surjective operator. Given у UH, the equation Ux = у has a unique solution of minimal norm, namely x = U^y.
The proof is identical with the proof of Theorem 1.5.2.
A.8 Some special functions
The Gaussian with parameter a > 0 is the function
pe(x) := e~ax2, x e R.
It plays a special role in Fourier analysis, partly because it is (up to constants) invariant under the Fourier transform:
Lemma A.8.1 For a > 0,
Another important function is the Mexican hat, which can be derived from the Gaussian with a = |:
Example A.8.2 We consider the Gaussian g(x) = e“ix2. Its first derivatives are
g'(x) = —xe~ix , g"(x) = —(1 — x2)e~^x .
One can show that l^,,(cc)|2^cc = I71"1/2; normalizing g" in L2(K) gives the Mexican hat
'ф(х) := -^=7T-1/4(1 - x2)e~ix .
The name of the function comes from its shape, see Figure A.l. In order to find the Fourier transform of we first use partial integration twice on
A.9 B-splines
413
the expression for Tgn'.
(•A/')(?) = [°° g"(x)e~21tilxdx J —oo
=	+ 2™7 Г 9'^e~2^dx
J — oo
= 27гг7Г[5(а;)е-27Г^]^“оо+27гп / g{x)e-2^x dx) \	J —oo	/
= - 4тг272р(7)
= — 47г2у/27Г72е-2’г 7 .
By the above calculation,
-0(7) = -^7г2л/2г7Г“1//472е_27га711
□
A.9 B-splines
In short, splines are functions which are piecewise polynomials; in the onedimensional case, this means that one can split the domain of a spline into intervals in such a way that the function is a polynomial on each interval. The points where the function changes from one polynomial to another
414 Appendix A
polynomial are called knots. In the general setting no assumption on the knots are made, and one can also consider splines in more variables.
For our purpose, however, the most elementary splines, namely, B-splines, will suffice. They are defined inductively: the first is simply
Bi(x) = X[-|,|]W,	(A.13)
and, assuming that we have defined Bn for some n 6 N, the next is defined by a convolution:
Bn+i(x) = Bn * Bi(x) = [ Bn(x J —oo
= [ Bn(x-t)dt.	(A.14)
The functions Bn defined by (A.13) and (A.14) are called В-splines, and n is the order. See Figures A.2, A.3 and A.4 for graphs of the first few B-splines. We collect some of their fundamental properties:
Theorem A.9.1 Given n E N, Bn has the following properties:
(i) Ifn > 2, then Bn e Cn“2(R).
(ii) supp Bn = [~|, and Bn > 0 on ] -	|[.
Bn(x)dx = 1 and	— k) = 1 for all x € R.
(iv) For any continuous function f : R —> C,
/ Bn(x)f(x)dx = / f(Xi H--------------h xn)dx1 • • • dxn. (A.15)
J-oo	J[-|,|]n
If n is even, the restriction of Bn to each interval [k,k + 1], к G Z, is a polynomial of degree at mostn — 1; ifn is odd, the restriction of Bn to each interval [к — |, к + |], к 6 Z, is a polynomial of degree at most n — 1.
Theorem A.9.1 can be proved by induction. Via (A.15) we can find the Fourier transform of Bn:
Corollary A.9.2
/ pTri'Y — g—irisy \ n	/ gi j-. /-Tr^y'j \ n
= (-—5—-—) =	.	(A.16)
\	27TZ7 J \ 7Г7 J
Proof.
jrBn(7) = Г Bn{xy-2^dx= [	e~2,ri<-X1+-+x^dxi---dxn
J-oo
(1	\ n	. 4 n
Г 2	\	/ р'К'И _
/ e-27r^drr = (	- ----- ) .	□
J \	27Г27	/
А.9 B-splines 415
—=b-
0.8;
0.6;
0.4;
0.2-
Г“
-2
Figure A.2. The B-spline B\.
With our definition of the B-splines all the functions Bn have support on a symmetric interval around zero. By Theorem A.9.1, the translated spline
T1
Nn(x) := Г»В„(х) = Bn(x - -)
has support on the interval [0, n]. The splines Nn are called cardinal B-splines; alternatively, they can be defined inductively exactly as the B-splines, starting with the function Ni = X[o,i]-
416
Appendix A
Using Corollary A.9.2 we can find the Fourier transform of the cardinal spline:

-ят7 ( 81П(7Г7)
\ TT7
(A.17)
We refer to the books by Chui [87] and de Boor [30] for information about general splines.
A. 10 Notes
In this Section we provide references for further reading.
Chapter 1: Frames in finite-dimensional spaces have recently attracted more attention because of their use in signal processing. See [18] by Benedetto and Fickus, as well as [59], [60] and [61] by Casazza and Kovacevic respectively Leon, Tremain. Frames {fk}^i where the frame condition even holds if some of the terms Ц/, A)|2 are replaced by — |</,A>|2 are studied by Peng and Waldron in [234], with special emphasis on finite frames. Results on tight frames and applications to coding and communication are given by Strohmer and Heath [269].
A. 10 Notes
417
Chapter 3: Donoho proved in [118] that the use of unconditional bases leads to optimal sparsity in signal representations. A survey on local trigonometric bases is in [28]. He and Volkmer use Riesz bases in the context of Sturm-Liouville equations in [167]. Local trigonometric bases were introduced by Malvar [220] and Coifman & Meyer [100].
Chapter 5:	Frames and operator algebras are studied by Han and Larson [165]. An algorithm (the matching pursuit algorithm) to represent elements in a Hilbert space via an overcomplete system was proposed by Mallat and Zhang in [219]; it applies to highly overcomplete systems which are not Bessel sequences, for example combined wavelet and Gabor systems with arbitrary parameters. See also [64]. There is a large literature on greedy algorithms and m-term approximation; see e.g, [276] by Temlyakov, where overcomplete systems are used. Christensen considered in [65] generalizations of frames, where the Bessel assumption is removed; the theme was taken up again in [52]. Ridgelets were introduced by Candes [41] as a tool to obtain better performance in image processing with images having edges. A survey on frame theory which discusses several open problem is given by Casazza [42]. Frames in Bargmann spaces are considered by Daubechies and Grossmann [107], by Grochenig and Walnut [160], and by Lyubarskii [216].
Chapter 6:	Casazza and Christensen proved in [51] that a frame is unconditional if and only if it is a near-Riesz basis.
Chapter 7:	Results on Riesz-Fischer sequences of exponentials are given by Reid in [240]. Estimates for the lower frame bound for a finite sequence of exponentials are given by Christensen and Lindner in [81] and extended to Gabor systems and wavelet systems in [82].
Chapter 8-10: Gabardo and Han [144] considered Gabor frames for subspaces of L2(JR) and operator algebras. A version of the Walnut representation of the frame operator in the Fourier domain can be found in [285].
Chapters 11—14: An early approach to wavelet frames in L2(R) is given by Frazier and Jawerth [141]. Libraries of frames, i.e., wavelet packet frames, are studied by Long and Chen in [215] and by Chen in [62]; a less advanced approach is in [75]. Wavelet frames in Sobolev are considered by Oswald [230]. Gribonval and Nielsen [148] use spline-generated wavelet frames in approximation theory. Among the alternatives to wavelet bases we mention brushlets, which were introduced by Laeng [203] and Meyer, Coifman [221]; they are based on a local trigonmetric basis multiplied with a bell function.
Chapter 17: Atomic decompositions of Hardy spaces appeared already in [101] by Coifman and Weiss.
418 Appendix A
In Chapter 17, R2 was extended to the Heisenberg group in order to be able to define a group representation. Alternatively, one can generalize the concept of a group representation. The operators p(a, b) = ЕаТь constitute a projective representation; a direct approach to atomic decomposition via projective representations is given by Christensen in [69]. Wilson bases in coorbit spaces are discussed by Feichtinger, Grochenig and Walnut in [132].
List of symbols
R :	The real numbers.
R+ : N :	The strictly positive real numbers. The natural numbers: 1,2,3,....
Z :	The integers.
Q:	The rational numbers.
C :	The complex numbers.
9cd(p,q) :	The largest common divisor for p, q G N.
Ы :	The integer part of x G IR, i.e., the largest integer not exceeding x.
x : X,Y : H,K. : £P(R) :	The complex conjugated of x G C. Banach spaces. Hilbert spaces. The space of measurable functions f : К C for which Jr lf<A)lPdx < 00.
C"=(R) :	The space of к times differentiable functions with a continuous /с-th derivative.
^/(7) = /(7) :	The Fourier transform, for f G L1(R) given by
£2(7) :	/(7) = Jr f(x)e-2™Tdx The space of square summable sequences on I.
И1:	The Lebesgue measure of a Borel set Z, or when I is discrete the number of elements in I.
xa ••	The indicator function for a set A, Xa(x) = 1 if x G A, otherwise 0.
A : A1 :	The closure of a set A. The orthogonal complement of a subset A in a Hilbert space
420 List of symbols
supp/ :	The support of the function f: supp/ = {t e IR : f(x) ф 0
6^^ :	The Kronecker delta: = 1 if к = j, 6k, j = 0 if к 0 j.
Ta : The translation operator (Taf)(x) = f(x — a)
Еь : The modulation operator (Ebf)(x) = e2lvlbx f(x)
Da : The dilation operator (Ра/)(т) = -i=/(^), a > 0
E> : The dilation operator (P/)(a?) = 21'/2f(2x).
S :	The	frame operator
T :	The	pre-frame operator
[P : The pseudo-inverse of the operator U
Mu :	The	kernel of the operator	U
1Zu •	The	range of the operator	U
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Index
l<j-independent, 46	4
(7-compact, 384
{ek +efc+1}, 98, 119, 146, 356 ax + b group, 386
absolutely convergent series, 37 adjoint matrix, 404 adjoint operator, 4, 408 admissibility condition, 251 affine group, 386 analysis operator, 4, 90 analyzing atom, 387 approximation order, 339 axiom of choice, 127
B-spline, 414
Balian-Low theorem, 82
Banach space, 36
Banach-Steinhaus theorem, 407 basis, 46 basis constant, 49 basis with brackets, 134 Battle-Lemarie wavelets, 75 Bessel bound, 52
Bessel sequence, 52
Beurling densities, 139 bi-frames, 277
biorthogonal MRA, 85
biorthogonal system, 50, 54
bounded operator, 407
canonical basis for £2(N), 58
canonical tight frame, 94
Cauchy sequence, 36
Cauchy-Schwarz’ inequality, 38, 404
Chebyshev acceleration, 13
coefficient functional, 49
coherent frame, 137
coherent structure, 72
commutator relations, 42
compact operator, 408
complete sequence, 37
complex exponentials, 29
compression, 84
condition (A), 202
condition (CC), 183
condition (R), 203, 233
condition (UCC), 203
condition number, 10
continuous frame, 115
continuous operator, 407
continuous wavelet transform, 251
convergent series, 36
convolution, 389, 414
438 Index
coorbit space, 394
dense subset, 37
dilation operator, 41
dilation parameter, 253
direct sum of Hilbert spaces, 406
discrete Fourier transform basis, 20
discrete Gabor system, 236
dual Banach space, 407
dual frame, 112
dual lattice, 202
dual of Gabor frame, 208
dual of wavelet frame, 274
dual Riesz basis, 64
dual window, 209
dyadic wavelet frame, 273
equivalence, <$-, 279
equivalence, r-, 279
Euler’s formulas, 72
exact frame, 88
expansion property, 79
exponential decay, 209
Fatou’s lemma, 405
Feichtinger algebra, 202
filter bank, 246
finite sequence, 37
Fourier frame, 157
Fourier series, 70
Fourier transform, 40
frame, 3, 88
frame algorithm, 11
frame bounds, 3, 88
frame coefficients, 7, 92
frame decomposition, 91
frame of exponentials, 157
frame of translates, 141
frame operator, 4, 90
frame radius, 158
frame wavelet set, 278
Fredholm operator, 358
functional, 407
Gabor basis, 72
Gabor frame, 172
Gabor frame set, 192
Gabor system, 32
Gabor transform, 169
gap, 357
Gaussian, 412
generalized dual, 121
generalized frame, 115
generalized inverse, 23, 28
generator, 172
Gram matrix, 60, 66
group representation, 384
Haar function, 73
Heisenberg group, 385
Hilbert space, 38
hyperbolic secant, 191
index of operator, 357
inner product, 403
integer oversampled Gabor system
378
integer part, 419
inversion formula, 40
irregular frame of translates, 141
irregular Gabor frame, 172
irregular Gabor system, 226
irregular wavelet frame, 267
Janssen representation, 212
Janssen’s tie, 191
Kadec’ 1/4-theorem, 163
kernel, 404
knots, 414
lattice, 172
Lebesgue point, 39, 42, 233
Lebesgue’s theorem, 405
left Haar measure, 384
linear independence, 2, 29, 229
localized frame, 396
locally compact group, 384
matching pursuit algorithm, 417
matrix representation, 404
Mexican hat, 263, 412
modulation operator, 41, 167, 235
modulation spaces, 394
moment problem, 134
Moore-Penrose inverse, 25
multi-window Gabor frame, 183
multiresolution analysis, 73
Index 439
multiwavelet frame, 274 multiwavelets, 85
nested subspaces, 74
Neumann’s theorem, 407
noise, 7, 117
nonexact frame, 96
nonharmonic Fourier series, 87, 158 normalized, 3
null-set, 405
oblique extension principle, 329 operator norm, 407
optimal frame bounds, 3, 88
optimal Riesz bounds, 68 orthogonal complement, 404 orthogonal projection, 409 orthonormal basis, 3, 56 orthonormalization trick, 152 overcomplete frame, 4, 96 oversampled filter bank, 247 oversampling of wavelet frame, 270
pair of dual wavelet frames, 277 paraunitary filter bank, 247 Parseval’s equation, 57 partition of unity, 391
Plancherel’s equation, 40 polar decomposition, 410 polarization identity, 409 polyphase matrix, 247 polyphase representation, 247 positive operator, 410 pre-frame operator, 4, 90 projective representation, 418 proper representation, 385 pseudo-inverse matrix, 23 pseudo-inverse operator, 93, 411 pseudodifferential operator, 231
quantization, 117
range, 405
rank, 15, 408
rational oversampled Gabor system, 218
rational oversampling, 210 redundant frame, 4, 96
refinable function, 288
refinement equation, 288
regular Gabor frame, 172
relatively separated set, 138
representation coefficient, 387
reproducing kernel, 157
reproducing kernel Hilbert space, 157
Riesz bounds, 68
Riesz frame, 126
Riesz sequence, 68
Riesz’ representation theorem, 408
Riesz’ subsequence theorem, 39
Riesz-Fischer sequence, 135
right-inverse operator, 410
Ron-Shen duality principle, 207
sampling, 236
sampling problem, 156
scaling parameter, 253
Schatten-von Neumann class, 231
Schauder basis, 46
Schrodinger representation, 385
Schur’s lemma, 62
self-adjoint matrix, 405
self-adjoint operator, 409
semiorthogonal, 310
separable normed space, 37
separated set, 138
separation constant, 138
set of sampling, 157
Shannon’s sampling theorem, 156
shift-invariant, 157
shift-invariant spaces, 397
shift-invariant systems, 192, 245
short-time Fourier transform, 169
sibling frames, 277
signal processing, 117, 246
signal transmission, 7, 117
sinc-function, 156
singular value, 230
singular value decomposition, 26
Sobolev space, 406
spectral theorem, 405
spline, 413
spline wavelet, 75
square root of positive operator, 410
stability, 347
strong operator topology, 407
subframe property, 128
symmetry group, 384
440
Index
synthesis operator, 4, 90
thresholding, 84
tight frame, 5, 88
time-frequency lattice, 202
time-frequency shifts, 169
torus, 384
total sequence, 37
translation operator, 41, 167, 236
translation parameter, 253
trigonometric polynomial, 31, 72
truncation, 117
two-scale symbol, 289
unconditional basis, 46
unconditional basis constant, 49
unconditionally convergent series, 3 uniform boundedness principle, 407 uniform density, 158
unimodular group, 384
unitary extension principle, 321
unitary operator, 408
Vandermonde matrix, 22, 30
vanishing moments, 83
Walnut representation, 204
wavelet, 32, 72, 250
wavelet basis, 72
wavelet frame, 253
wavelet structure, 250
wavelet system, 250
Weierstrass’ theorem, 28
Wexler-Raz theorem, 210
Weyl correspondence, 231
Weyl-Heisenberg frame, 172
Wiener space, 187
Wigner distribution, 231
Wilson basis, 232
window function, 169, 172
Zak transform, 215
Zibulski-Zeevi matrix, 218
Zorn’s lemma, 127