Applied and Numerical Harmonic Analysis

Series EditorJohn J. BenedettoUniversity of MarylandCollege Park, MD, USA

Editorial Advisory Board

Akram AldroubiVanderbilt UniversityNashville, TN, USA

Douglas CochranArizona State UniversityPhoenix, AZ, USA

Hans G. FeichtingerUniversity of ViennaVienna, Austria

Christopher HeilGeorgia Institute of TechnologyAtlanta, GA, USA

Stephane JaffardUniversity of Paris XIIParis, France

Jelena KovacevicCarnegie Mellon UniversityPittsburgh, PA, USA

Gitta KutyniokTechnische Universitat BerlinBerlin, Germany

Mauro MaggioniDuke UniversityDurham, NC, USA

Zuowei ShenNational University of SingaporeSingapore, Singapore

Thomas StrohmerUniversity of CaliforniaDavis, CA, USA

Yang WangMichigan State UniversityEast Lansing, MI, USA

More information about this series at http://www.springer.com/series/4968

Ole Christensen

An Introduction to Framesand Riesz Bases

Second Edition

Ole ChristensenDepartment of Mathematics

and Computer ScienceTechnical University of DenmarkLyngby, Denmark

ISSN 2296-5009 ISSN 2296-5017 (electronic)Applied and Numerical Harmonic AnalysisISBN 978-3-319-25611-5 ISBN 978-3-319-25613-9 (eBook)DOI 10.1007/978-3-319-25613-9

Library of Congress Control Number: 2015954170

Mathematics Subject Classification (2010): 41-01, 41-02, 42-01, 42-02, 42C15, 42C40

© Springer Science+Business Media New York 2003© Springer International Publishing Switzerland 2016This work is subject to copyright. All rights are reserved by the Publisher, whether the whole or partof the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations,recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission orinformation storage and retrieval, electronic adaptation, computer software, or by similar or dissimilarmethodology now known or hereafter developed.The use of general descriptive names, registered names, trademarks, service marks, etc. in this publi-cation does not imply, even in the absence of a specific statement, that such names are exempt fromthe relevant protective laws and regulations and therefore free for general use.The publisher, the authors and the editors are safe to assume that the advice and information in thisbook are believed to be true and accurate at the date of publication. Neither the publisher nor theauthors or the editors give a warranty, express or implied, with respect to the material contained hereinor for any errors or omissions that may have been made.

Printed on acid-free paper

This Springer imprint is published by Springer NatureThe registered company is Springer International Publishing AG Switzerland

To Khadija, Jakob, Sara; and Karen

ANHA Series Preface

The Applied and Numerical Harmonic Analysis (ANHA) book series aimsto provide the engineering, mathematical, and scientific communities withsignificant developments in harmonic analysis, ranging from abstract har-monic analysis to basic applications. The title of the series reflects theimportance of applications and numerical implementation, but richnessand relevance of applications and implementation depend fundamentallyon the structure and depth of theoretical underpinnings. Thus, from ourpoint of view, the interleaving of theory and applications and their creativesymbiotic evolution is axiomatic.Harmonic analysis is a wellspring of ideas and applicability that has flour-

ished, developed, and deepened over time within many disciplines and bymeans of creative cross-fertilization with diverse areas. The intricate andfundamental relationship between harmonic analysis and fields such as sig-nal processing, partial differential equations (PDEs), and image processingis 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, time-frequency analysis, and fractal geometry, as well as the diverse topics thatimpinge on them.For example, wavelet theory can be considered an appropriate tool to

deal with some basic problems in digital signal processing, speech and

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viii ANHA Series Preface

image processing, geophysics, pattern recognition, biomedical engineering,and turbulence. These areas implement the latest technology from sam-pling methods on surfaces to fast algorithms and computer vision methods.The underlying mathematics of wavelet theory depends not only on clas-sical Fourier analysis but also on ideas from abstract harmonic analysis,including von Neumann algebras and the affine group. This leads to astudy of the Heisenberg group and its relationship to Gabor systems and ofthe metaplectic group for a meaningful interaction of signal decompositionmethods.The unifying influence of wavelet theory in the aforementioned topics

illustrates the justification for providing a means for centralizing and dis-seminating information from the broader, but still focused, area of harmonicanalysis. This will be a key role of ANHA. We intend to publish with thescope 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 commit-ment to publish major advances in applicable topics such as the following,where harmonic analysis plays a substantial role:

Biomathematics, bioengineering,and biomedical signal processing;Communications and RADAR;Compressive sensing (sampling)

and sparse representations;Data science, data mining,and dimension reduction;

Fast algorithms;Frame theory and noise reduction;

Image processingand super-resolution;

Machine learning;Phaseless reconstruction;Quantum informatics;

Remote sensing;Sampling theory;

Spectral estimation;Time-frequency and Time-scale

analysis—Gabor theoryand Wavelet theory

The above point of view for the ANHA book series is inspired by thehistory 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 engineeringand scientific phenomena, and on the solution of some of the most impor-tant problems in mathematics and the sciences. Historically, Fourier serieswere developed in the analysis of some of the classical PDEs of mathe-matical physics; these series were used to solve such equations. In order tounderstand Fourier series and the kinds of solutions they could represent,some of the most basic notions of analysis were defined, e.g., the conceptof “function.” Since the coefficients of Fourier series are integrals, it is no

ANHA Series Preface ix

surprise that Riemann integrals were conceived to deal with uniquenessproperties of trigonometric series. Cantor’s set theory was also developedbecause of such uniqueness questions.A basic problem in Fourier analysis is to show how complicated phe-

nomena, such as sound waves, can be described in terms of elementaryharmonics. There are two aspects of this problem: first, to find, or evendefine properly, the harmonics or spectrum of a given phenomenon, e.g.,the spectroscopy problem in optics; second, to determine which phenomenacan 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 sciences. For example, Wiener’s Tauberiantheorem in Fourier analysis not only characterizes the behavior of the primenumbers but also provides the proper notion of spectrum for phenomenasuch as white light; this latter process leads to the Fourier analysis asso-ciated with correlation functions in filtering and prediction problems, andthese problems, in turn, deal naturally with Hardy spaces in the theory ofcomplex variables.Nowadays, some of the theory of PDEs has given way to the study

of Fourier integral operators. Problems in antenna theory are stud-ied in terms of unimodular trigonometric polynomials. Applications ofFourier analysis abound in signal processing, whether with the fast Fouriertransform (FFT), or filter design, or the adaptive modeling inherent intime-frequency-scale methods such as wavelet theory.The coherent states of mathematical physics are translated and mod-

ulated Fourier transforms, and these are used, in conjunction with theuncertainty principle, for dealing with signal reconstruction in com-munications theory. We are back to the raison d’etre of the ANHAseries!

College Park, MD, USA John J. Benedetto

Preface to the First Edition

Frames have fascinated me since day one. Every student in mathematicslearns about bases in vector spaces, allowing one to represent each elementin a convenient and unique way. One day in 1990, Henrik Stetkær, who wasmy master’s thesis advisor, showed me the definition of a frame and told methat a frame is some kind of “overcomplete basis”: one can also representeach element in the vector space via a frame, but the representation mightnot 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?Why did the idea that overcompleteness by itself could be useful nevercame up?A search on Mathematical Reviews or Zentralblatt shows only a few titles

of books or articles concerning frames published before 1991; among thosewe mention the original paper by Duffin and Schaeffer [262], the excellentbook by Young [622], and the important papers by Daubechies, Grossmann,and Meyer [244], Daubechies [241], and Heil and Walnut [395]. Now, justten years later, hundreds of papers have the word frame in the title, andperhaps a thousand discuss one or more results. Today, no single book cantreat 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 agraduate course, as well as by professional readers. For use in a graduatecourse, a number of exercises is included; they appear at the end of each

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chapter. The number of exercises gives a hint of the level of the chapter:there are many exercises in the introductory chapters, but only few in theadvanced chapters. In the same spirit, almost all results in the introductorychapters appear with full proofs; in the advanced chapters several resultsare presented without proofs. We believe it is more useful to state a largenumber of results in a common framework than to see detailed proofs ofsignificantly fewer statements; this feature also makes the book useful as areference.The content can be split naturally into three parts: Chapters 1–7 de-

scribe the theory on an abstract level, Chapters 9–18 describe explicitconstructions in L2-spaces, and Chapters 22–24 deal with selected researchtopics.In Chapters 1–7 almost all results concern frames in general Hilbert

spaces. The goal is an almost complete treatment of the known results forframes. For the explicit constructions in L2(−π, π) and L2(R), which ap-pear in Chapters 9–18, the situation is different. For this part, I was forcedto concentrate on selected parts of the theory. Since we are mainly inter-ested in overcomplete systems, the theory presented in these chapters ispart of a larger picture, and the reader will certainly benefit from knowl-edge of the background. Chapter 9 connects to the theory for nonharmonicFourier series, cf. the book [622] by Young. Gabor frames arise naturallyin the context of time-frequency analysis, and the book by Grochenig [340]will clarify the role of Chapters 11–13 in time-frequency analysis. Finally,the role of wavelets is highlighted in the classic book [242] by Daubechies,which also gives the motivation for the study of frames arising from mul-tiscale methods in Chapters 17–18. Technically, we do not rely on any ofthese books (only at a few places will we refer to results from them withoutproof), but they put the results of frame studies in the right perspective.Chapters 9–18 are also influenced by the fact that the material is used inseveral areas of applied mathematics; the reader will observe that althoughthis is a book about mathematics, those chapters concentrate on applicableways to construct frames rather than on abstract characterizations.Let us describe the chapters in more detail. Chapter 1 presents basic re-

sults in finite-dimensional vector spaces with an inner product. This enablesa reader with a basic knowledge of linear algebra to understand the ideabehind frames without the technical complications in infinite-dimensionalspaces. Many of the topics from the rest of the book are presented here, soChapter 1 can also serve as an introduction to the later chapters.Chapter 2 collects some definitions and conventions concerning infinite-

dimensional vector spaces. Special attention is given to the Hilbert spaceL2(R) and operators hereon. We expect the reader to be familiar with thismaterial; the chapter is too short to be considered as an introduction tothe subject.

Preface xiii

Chapter 3 describes the classical theory for bases in Hilbert spaces andBanach theory. The examples in this chapter are chosen so they leadnaturally to the constructions in Chapters 9–18.Chapter 4 highlights some of the limitations on the properties one

can obtain from bases, and thus provides motivation for consideringgeneralizations of bases.Chapters 5–7 contain the core material about frames and Riesz bases.

Chapter 5 is classical, but Chapter 7 contains several results published inthe last five years. The interplay between frames and bases is discussed indetail in Chapter 7, and we also discuss frames that become bases when acertain number of elements are deleted.Chapters 9–18 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(−π, π)and frames of translates in L2(R), are discussed in Chapter 9. If one wantsto consider frames in L2(R), these frames easily lead to Gabor frames,which is the subject of Chapters 11–14. Wavelet frames are introduced inChapter 15, and sufficient conditions to find them are given for arbitrarydilation parameter a > 1 and translation parameter b > 0. Some resultsconcerning irregular wavelet frames are also presented there. Chapter 16specializes to the important case a = 2, b = 1, which has attracted muchattention during the past ten years. Constructions via multiscale methodsare the focus in Chapters 17–18.In Chapter 22, 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 arenever completely exact, this question is very important for applications.Chapter 23 presents methods for the approximation of the inverse frame

operator using finite subsets of the frame. Since every application of frametheory has to be performed in a finite-dimensional vector space, this topicis also of fundamental importance for applications.Chapter 24 deals with extensions and generalizations of the material

from the previous chapters. Expansions in Banach spaces and their re-lationship to frames in Hilbert spaces are discussed, as well as framesappearing via integrable group representations. The latter subject givesa unified description of the frames from Chapters 11–15.Finally, an Appendix collects several basic results for easy reference. It

also contains material on pseudo-inverse operators and splines which is notexpected 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 partsof Chapters 2–3. Chapter 4 is important as motivation, and Chapter 5 isalso core material. But after covering these chapters, a course can continuein several ways. One possibility is to follow a theoretical track and considerthe relationship between frames and bases in more detail; this could befollowed by a study of one of the three final chapters. Another possibility is

xiv Preface

to continue with constructions of exponential frames and Gabor frames, orwavelet frames. If wavelets are chosen as the subject, it is worth noticingthat 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 arevery useful in several areas of applied mathematics, including signal pro-cessing and image processing. But this part of the story should be told bythe people who are directly involved in it, and we will only sketch a fewapplications.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 pagesof notes, which were written jointly with Torben Klint Jensen, who wasat that time a master’s student. My original idea was to write a bookconcentrating on frames in general Hilbert spaces; I am very happy thatThomas Strohmer and an anonymous reviewer suggested that I further gointo detail with wavelet and Gabor systems. Their ideas added more thana hundred pages to the book and extended the scope considerably. Veryuseful 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 Stetkær, and Diana Stoeva read partsof 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 Uni-

versity of Denmark for providing me with the excellent working conditionsthat made it possible to concentrate on the book for two semesters. Inaddition, a large part of the book was written during a stay at the re-search group NuHAG at the University of Vienna. It is a pleasure to thankthe group leader, Hans Feichtinger, and the members of NuHAG for theirsupport.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 Kostant,for their assistance and support. Thanks are also given to Elizabeth Loewfrom Texniques, who helped with Latex problems.Finally, I acknowledge support from the WAVE-program, sponsored by

the Danish Science Foundation.

Ole ChristensenKgs. Lyngby, DenmarkSeptember 2002

Preface to the Second Edition

As I wrote the first edition of the book during 2001/2002, one of thegoals was that at least the list of references should contain most of theframe literature. Now, 14 years later, even this very modest goal cannotbe reached anymore. During the last 20 years frames have experienced anever-increasing popularity, and they show up in many different contexts,explicitly or implicitly. Considering just four of the key topics, namely,(i) “Frames in finite-dimensional spaces,” (ii) “Frames in general Hilbertspaces,” (iii) “Frames in Gabor analysis,” and (iv) “Frames and waveletanalysis,” each of these topics could easily fill a book of the same size asthe current book. Therefore one of the major decisions during the work onthe second edition has been what to include – and at what level of details.My choice has been to follow the line from the first edition and present thecore material (and frequently less known material that should belong to thecore) in great detail, while other topics are treated as research topics withmore focus on the connections between the results than the proofs. Thefact that many recent and advanced results are presented without proofsmade it possible still to give a quite broad picture of the frame theory; butclearly it also leaves a gap open for other authors who would like to give adetailed presentation focusing on one of the topics.The new material mainly occurs in new chapters and sections, but of

course the entire book has been updated with additional results and com-ments. On very compressed form the main additions can be described asfollows:

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• Sections 1.2 on tight frames and dual pairs of frames in finite-dimensional spaces. Section 1.8 on fusion frames. Section 1.9 onapplications of frames. Finally, Section 1.10 which relates the prop-erties of the harmonic frames to the ongoing research within finiteframe theory.

• Extension and rearrangement of Chapter 2. Many results from theformer appendix now appear here.

• Section 3.7, a new section on Riesz sequences; and Section 3.10 onsampling an analog-digital conversion.

• Several updates and additions in Chapter 4, which motivate the stepfrom bases to frames.

• Section 5.2, a new section on frame sequences.

• Chapter 6, a new chapter that collects results about tight frames anddual pairs of frames in general Hilbert spaces.

• Section 7.2, a new section about relations between frames and theirsubsequences, with focus on the “strange” behavior of the lower framebounds for finite subfamilies. And Section 7.7, a short section on theFeichtinger conjecture.

• Chapter 8, a new chapter on selected topics in general frame theory.It contains sections on G-frames, localization of frames, the R-dualsequences, a frame-like theory via unbounded operators, as well as adiscussion of frames in the context of signal processing.

• Section 9.4, Section 9.5, and Section 9.7: new sections about thecanonical dual of a frame of translates and oblique duals, as wellas applications of frames of translates within sampling theory.

• Chapter 10, a new chapter on shift-invariant systems (parts ofthe material previously appeared within the presentation of Gaborframes).

• Extensions and updates in Section 11.6 on Gabor frames gener-ated by special functions. Section 11.7, a new section collecting theknown connections between B-splines and Gabor frames, as well asdiscussions about open problems.

• Chapter 12, a new chapter on dual pairs of Gabor frames and tightGabor frames.

• Section 13.1, a new section about the duality principle. Section 13.5,a new section about localized Gabor frames. And Section 13.7, a newsection on time-frequency localization.

• Chapter 14: new sections concerning duality of Gabor frames in �2(Z)and �2(Zd), and explicit construction of such frames based on dual

Preface xvii

pairs of Gabor frames for L2(R). Construction of periodic Gaborframes in L2(0, L), and description of the transition from a Gaborsystem in L2(R) to a finite-dimensional model in C

L.

• Section 15.3, a new section on dual pairs of wavelet frames.

• Chapter 19, a new chapter on selected topics on wavelet frames. Someof the sections also appeared in the first edition of the book, but thesections on the extension problem and signal processing are new.

• Chapter 20, a new chapter on generalized shift-invariant systems.

• Chapter 21, a new chapter on frames on locally compact abeliangroups.

• Section 23.3, a new section that yields convergence estimates in thecontext of finite-dimensional approximations of the inverse frameoperator.

• Chapter 24 on Banach frames: the entire chapter has been updatedwith more recent results.

• Section A.5 and Section A.6, new sections stating the key propertiesof the modulation spaces and the Feichtinger algebra. Section A.9and Section A.10, new sections on exponential B-splines and splineson LCA groups.

I would like to thank all the friends, colleagues, and students who havecontributed to the current second edition. First and foremost I would like tothank my coauthors, who have definitely inspired me and shaped my viewand understanding of frames over the years. Many of the papers with mycoauthors were used as the starting point for various sections and chapters.For example, my papers with Hong Oh Kim and Rae Young Kim form thebasis for Sections 6.4, 11.7, 12.5, 12.6, and 12.7; similarly, the paper [176]with Say Song Goh was the driving force behind most of the sections inChapter 21.I would also like to thank Hong Oh Kim and Rae Young Kim for organiz-

ing and supporting about 20 visits to Korea Advanced Institute for Scienceand Technology (KAIST) over the years, and for the many pleasant hourswe spend working on joint problems; and Say Song Goh, whose many invi-tations to National University of Singapore (NUS) also gave me scientificinspiration and excellent working conditions, with direct influence on thecurrent book.It is a great pleasure to thank Henrik Stetkær, who used the first edition

as textbook in several master courses at the University of Aarhus; this ledto the discovery of several misprints and imprecisions, which I have triedto correct.I thank Jakob Lemvig and Mads Sielemann Jakobsen for giving me access

to a note providing a direct proof of the duality principle in Gabor analysis.

xviii Preface

During the preparation of the manuscript, I got help from many col-leagues and students to spot typing mistakes, bad formulations, etc.;I thank Say Song Goh, Marzieh Hasannasab, Christina Hildebrandt,Mads Sielemann Jakobsen, Jakob Lemvig, Diana Stoeva, and Jordy vanVelthoven for their help, which clearly improved the manuscript.Finally I want to thank John Benedetto for his never-ending support and

positive attitude. I also thank the staff at Birkhauser, especially DanielleWalker, for their support during the entire process.

Ole ChristensenKgs. Lyngby, DenmarkAugust 2015

Contents

Preface xi

1 Frames in Finite-DimensionalInner Product Spaces 11.1 Some Basic Facts About Frames . . . . . . . . . . . . . 31.2 Extensions to Tight Frames and Dual Frames . . . . . . 101.3 Frame Bounds and Frame Algorithms . . . . . . . . . . . 121.4 Frames in C

n . . . . . . . . . . . . . . . . . . . . . . . . 161.5 Frames and the Discrete Fourier Transform . . . . . . . . 211.6 Pseudo-inverses and the Singular Value Decomposition . 281.7 Finite-Dimensional Function Spaces . . . . . . . . . . . . 331.8 Fusion Frames . . . . . . . . . . . . . . . . . . . . . . . . 371.9 Applications of Finite Frames . . . . . . . . . . . . . . . 381.10 Remarks on Recent Frame Constructions . . . . . . . . . 421.11 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . 44

2 Infinite-Dimensional Vector Spacesand Sequences 472.1 Banach Spaces and Sequences . . . . . . . . . . . . . . . 472.2 Operators on Banach Spaces . . . . . . . . . . . . . . . . 502.3 Hilbert Spaces . . . . . . . . . . . . . . . . . . . . . . . . 52

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2.4 Operators on Hilbert Spaces . . . . . . . . . . . . . . . . 532.5 The Pseudo-inverse Operator . . . . . . . . . . . . . . . 562.6 A Moment Problem . . . . . . . . . . . . . . . . . . . . . 582.7 The Spaces Lp(R), L2(R), �p(N), and �2(N) . . . . . . . 592.8 The Fourier Transform and Convolution . . . . . . . . . 612.9 Operators on L2(R) . . . . . . . . . . . . . . . . . . . . . 632.10 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . 65

3 Bases 673.1 Bases in Banach Spaces . . . . . . . . . . . . . . . . . . . 683.2 Bessel Sequences in Hilbert Spaces . . . . . . . . . . . . 733.3 Bases and Biorthogonal Systems in H . . . . . . . . . . . 763.4 Orthonormal Bases . . . . . . . . . . . . . . . . . . . . . 793.5 The Gram Matrix . . . . . . . . . . . . . . . . . . . . . . 823.6 Riesz Bases . . . . . . . . . . . . . . . . . . . . . . . . . 863.7 Riesz Sequences . . . . . . . . . . . . . . . . . . . . . . . 923.8 Fourier Series and Gabor Bases . . . . . . . . . . . . . . 943.9 Wavelet Bases . . . . . . . . . . . . . . . . . . . . . . . . 973.10 Sampling and Analog–Digital Conversion . . . . . . . . . 1023.11 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . 105

4 Bases and Their Limitations 1094.1 Bases and the Expansion Property . . . . . . . . . . . . 1094.2 Gabor Systems and the Balian–Low Theorem . . . . . . 1134.3 Bases and Wavelets . . . . . . . . . . . . . . . . . . . . . 1154.4 General Shortcomings . . . . . . . . . . . . . . . . . . . . 1184.5 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . 118

5 Frames in Hilbert Spaces 1195.1 Frames and Their Properties . . . . . . . . . . . . . . . . 1205.2 Frame Sequences . . . . . . . . . . . . . . . . . . . . . . 1275.3 Frames and Operators . . . . . . . . . . . . . . . . . . . 1285.4 Frames and Bases . . . . . . . . . . . . . . . . . . . . . . 1305.5 Characterization of Frames . . . . . . . . . . . . . . . . . 1365.6 Continuous Frames . . . . . . . . . . . . . . . . . . . . . 1455.7 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . 148

6 Tight Frames and Dual Frame Pairs 1536.1 Tight Frames . . . . . . . . . . . . . . . . . . . . . . . . 1546.2 Extension of Bessel Sequences to Tight Frames . . . . . 1556.3 The Dual Frames . . . . . . . . . . . . . . . . . . . . . . 1566.4 Extension Problems for Bessel Sequences . . . . . . . . . 1616.5 Approximately Dual Frames . . . . . . . . . . . . . . . . 1626.6 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . 163

Contents xxi

7 Frames Versus Riesz Bases 165

7.1 Conditions for a Frame Being a Riesz Basis . . . . . . . 166

7.2 Frames and Their Subsequences . . . . . . . . . . . . . . 168

7.3 Riesz Frames and Near-Riesz Bases . . . . . . . . . . . . 170

7.4 Frames Containing a Riesz Basis . . . . . . . . . . . . . . 171

7.5 A Frame Which Does Not Contain a Basis . . . . . . . . 173

7.6 A Moment Problem . . . . . . . . . . . . . . . . . . . . . 179

7.7 The Feichtinger Conjecture . . . . . . . . . . . . . . . . . 181

7.8 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . 181

8 Selected Topics in Frame Theory 183

8.1 G-Frames . . . . . . . . . . . . . . . . . . . . . . . . . . . 184

8.2 Localization of Frames . . . . . . . . . . . . . . . . . . . 188

8.3 The R-Duals of a Frame . . . . . . . . . . . . . . . . . . 190

8.4 Frame Theory via Unbounded Operators . . . . . . . . . 194

8.5 Frames and Signal Processing . . . . . . . . . . . . . . . 196

8.6 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . 198

9 Frames of Translates 199

9.1 Sequences in Rd . . . . . . . . . . . . . . . . . . . . . . . 200

9.2 Frames of Translates . . . . . . . . . . . . . . . . . . . . 203

9.3 Frames of Integer-Translates . . . . . . . . . . . . . . . . 210

9.4 The Canonical Dual Frame . . . . . . . . . . . . . . . . . 213

9.5 Frames of Translates and Oblique Duals . . . . . . . . . 218

9.6 Irregular Frames of Translates . . . . . . . . . . . . . . . 226

9.7 Sampling Theory and Applications . . . . . . . . . . . . 229

9.8 Frames of Exponentials . . . . . . . . . . . . . . . . . . . 232

9.9 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . 238

10 Shift-Invariant Systems in L2(R) 241

10.1 Frame Properties of Shift-Invariant Systems . . . . . . . 241

10.2 Representations of the Frame Operator . . . . . . . . . . 253

10.3 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . 256

11 Gabor Frames in L2(R) 257

11.1 Continuous Representations . . . . . . . . . . . . . . . . 259

11.2 Gabor Frames {EmbTnag}m,n∈Z for L2(R) . . . . . . . . 262

11.3 Necessary Conditions . . . . . . . . . . . . . . . . . . . . 266

11.4 Sufficient Conditions . . . . . . . . . . . . . . . . . . . . 268

11.5 The Wiener Space W . . . . . . . . . . . . . . . . . . . . 275

11.6 The Frame Set and Special Functions . . . . . . . . . . . 279

11.7 Gabor Frames Generated by B-Splines . . . . . . . . . . 283

11.8 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . 285

xxii Contents

12 Gabor Frames and Duality 28712.1 Popular Gabor Conditions . . . . . . . . . . . . . . . . . 28812.2 Representations of the Gabor Frame Operator

and Duality . . . . . . . . . . . . . . . . . . . . . . . . . 28912.3 The Duals of a Gabor Frame . . . . . . . . . . . . . . . . 29512.4 The Canonical Dual Window . . . . . . . . . . . . . . . . 30212.5 Explicit Construction of Dual Frame Pairs . . . . . . . . 30612.6 Windows with Short Support and High Regularity . . . . 31012.7 Extension of Bessel Sequences to Dual Pairs . . . . . . . 31812.8 Approximately Dual Gabor Frames . . . . . . . . . . . . 31912.9 Tight Gabor frames . . . . . . . . . . . . . . . . . . . . . 32012.10 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . 324

13 Selected Topics on Gabor Frames 32713.1 The Duality Principle . . . . . . . . . . . . . . . . . . . . 32813.2 The Zak Transform . . . . . . . . . . . . . . . . . . . . . 33013.3 The Lattice Parameters . . . . . . . . . . . . . . . . . . . 33613.4 Irregular Gabor Systems . . . . . . . . . . . . . . . . . . 34013.5 Localized Gabor Frames . . . . . . . . . . . . . . . . . . 34513.6 Wilson Bases . . . . . . . . . . . . . . . . . . . . . . . . . 34713.7 Time–Frequency Localization of Gabor Expansions . . . 34813.8 Applications of Gabor Frames . . . . . . . . . . . . . . . 35413.9 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . 356

14 Gabor Frames in �2(Z), L2(0, L),CL 35914.1 Translation and Modulation on �2(Z) . . . . . . . . . . . 36014.2 Dual Gabor Frames in �2(Z) . . . . . . . . . . . . . . . . 36114.3 Dual Gabor Frames in �2(Z) Through Sampling . . . . . 36214.4 Discrete Gabor Frames Through Sampling . . . . . . . . 36514.5 Gabor Frames for L2(0, L) via Periodization . . . . . . . 37214.6 Gabor Frames in C

L . . . . . . . . . . . . . . . . . . . . 37514.7 Shift-Invariant Systems . . . . . . . . . . . . . . . . . . . 38014.8 Frames in �2(Z) and Filter Banks . . . . . . . . . . . . . 38114.9 Gabor frames in �2(Zd) . . . . . . . . . . . . . . . . . . . 38314.10 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . 383

15 General Wavelet Frames in L2(R) 38515.1 The Continuous Wavelet Transform . . . . . . . . . . . . 38715.2 Sufficient and Necessary Conditions . . . . . . . . . . . . 38915.3 Dual Pairs of Wavelet Frames . . . . . . . . . . . . . . . 40115.4 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . 405

16 Dyadic Wavelet Frames for L2(R) 40716.1 Wavelet Frames and Their Duals . . . . . . . . . . . . . 40816.2 Tight Wavelet Frames . . . . . . . . . . . . . . . . . . . . 411

Contents xxiii

16.3 Wavelet Frame Sets . . . . . . . . . . . . . . . . . . . . . 41216.4 Frames and Multiresolution Analysis . . . . . . . . . . . 41516.5 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . 415

17 Frame Multiresolution Analysis 41717.1 Frame Multiresolution Analysis . . . . . . . . . . . . . . 41817.2 Sufficient Conditions . . . . . . . . . . . . . . . . . . . . 41917.3 Relaxing the Conditions . . . . . . . . . . . . . . . . . . 42317.4 Construction of Frames . . . . . . . . . . . . . . . . . . . 42517.5 Frames with Two Generators . . . . . . . . . . . . . . . . 44217.6 Some Limitations . . . . . . . . . . . . . . . . . . . . . . 44417.7 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . 444

18 Wavelet Frames via Extension Principles 44518.1 The General Setup . . . . . . . . . . . . . . . . . . . . . 44618.2 The Unitary Extension Principle . . . . . . . . . . . . . . 44818.3 Applications to B-splines I . . . . . . . . . . . . . . . . . 45418.4 The Oblique Extension Principle . . . . . . . . . . . . . 45918.5 Fewer Generators . . . . . . . . . . . . . . . . . . . . . . 46318.6 Applications to B-splines II . . . . . . . . . . . . . . . . 46618.7 Approximation Orders . . . . . . . . . . . . . . . . . . . 47118.8 Construction of Pairs of Dual Wavelet Frames . . . . . . 47318.9 Applications to B-splines III . . . . . . . . . . . . . . . . 47618.10 The MRA Literature and Applications . . . . . . . . . . 47718.11 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . 478

19 Selected Topics on Wavelet Frames 47919.1 Irregular Wavelet Frames . . . . . . . . . . . . . . . . . . 48019.2 Oversampling of Wavelet Frames . . . . . . . . . . . . . 48219.3 An Open Extension Problem . . . . . . . . . . . . . . . . 48319.4 The Signal Processing Perspective . . . . . . . . . . . . . 48519.5 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . 492

20 Generalized Shift-Invariant Systems in L2(Rd) 49320.1 Analysis in R

d and Notation . . . . . . . . . . . . . . . . 49420.2 The Case of One Generator . . . . . . . . . . . . . . . . 49720.3 Frames with Multiple Generators . . . . . . . . . . . . . 50120.4 Dual Pairs of Frames with Multiple Generators . . . . . 50320.5 Gabor Systems in L2(Rd) . . . . . . . . . . . . . . . . . . 50720.6 Wavelet Systems in L2(Rd) . . . . . . . . . . . . . . . . . 51120.7 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . 516

21 Frames on Locally Compact Abelian Groups 51921.1 LCA Groups . . . . . . . . . . . . . . . . . . . . . . . . . 52121.2 Fourier Analysis on LCA Groups . . . . . . . . . . . . . 526

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21.3 Gabor Systems on LCA Groups . . . . . . . . . . . . . . 530

21.4 Basic Frame Calculations in L2(G). . . . . . . . . . . . . 532

21.5 Explicit Gabor Frame Constructions in L2(G) . . . . . . 53721.6 GSI Systems on LCA Groups . . . . . . . . . . . . . . . 54421.7 Generalized Translation-Invariant Systems . . . . . . . . 54821.8 Co-compact Gabor Systems . . . . . . . . . . . . . . . . 55421.9 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . 556

22 Perturbation of Frames 55722.1 A Paley–Wiener Theorem for Frames . . . . . . . . . . . 55822.2 Compact Perturbation . . . . . . . . . . . . . . . . . . . 56522.3 Perturbation of Frame Sequences . . . . . . . . . . . . . 56722.4 Perturbation of Gabor frames . . . . . . . . . . . . . . . 57022.5 Perturbation of Wavelet Frames . . . . . . . . . . . . . . 57322.6 Perturbation of the Haar Wavelet . . . . . . . . . . . . . 57422.7 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . 574

23 Approximation of the Inverse Frame Operator 57723.1 The First Approach . . . . . . . . . . . . . . . . . . . . . 57823.2 The Casazza–Christensen Method . . . . . . . . . . . . . 58223.3 Convergence Estimates for Localized Frames . . . . . . . 58923.4 Applications to Gabor Frames . . . . . . . . . . . . . . . 59123.5 Integer Oversampled Gabor Frames . . . . . . . . . . . . 59423.6 The Finite Section Method . . . . . . . . . . . . . . . . . 59523.7 The Finite Section Method for Gabor Frames . . . . . . 59823.8 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . 599

24 Expansions in Banach Spaces 60124.1 Representations of Locally Compact Groups . . . . . . . 60224.2 Feichtinger–Grochenig Theory . . . . . . . . . . . . . . . 60624.3 Banach Frames . . . . . . . . . . . . . . . . . . . . . . . 61124.4 p-frames . . . . . . . . . . . . . . . . . . . . . . . . . . . 61624.5 Gabor Systems and Wavelets in Lp(R) and Related

Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61924.6 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . 620

A Appendix 623A.1 Linear Algebra . . . . . . . . . . . . . . . . . . . . . . . . 623A.2 Integration . . . . . . . . . . . . . . . . . . . . . . . . . . 624A.3 Locally Compact Groups . . . . . . . . . . . . . . . . . . 625A.4 Some Infinite-Dimensional Vector Spaces . . . . . . . . . 626A.5 Modulation Spaces . . . . . . . . . . . . . . . . . . . . . 627A.6 Feichtinger’s algebra S0 . . . . . . . . . . . . . . . . . . . 629A.7 Some Special Functions . . . . . . . . . . . . . . . . . . . 632

Contents xxv

A.8 B-Splines . . . . . . . . . . . . . . . . . . . . . . . . . . . 633A.9 Exponential B-Splines . . . . . . . . . . . . . . . . . . . 639A.10 Splines on Locally Compact Abelian Groups . . . . . . . 641A.11 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 643

List of Symbols 645

References 647

Index 691