# S. Axler Gehring K.A. Ribet

## Transcript Of S. Axler Gehring K.A. Ribet

Universitext

Editorial Board (North America):

S. Axler F.W. Gehring

K.A. Ribet

Springer Science+Business Media, LLC

Universitext

Editors (North America): S. Axler, F.W. Gehring, and K.A. Ribet

Aksoy/Khamsi: Nonstandard Methods in Fixed Point Theory Andersson: Topics in Complex Analysis Aupetit: A Primer on Spectral Theory BaclunanlNaricilBeckenstein: Fourier and Wavelet Analysis Badescu: Algebraic Surfaces BalakrishnanlRanganathan: A Textbook of Graph Theory Balser: Formal Power Series and Linear Systems of Meromorphic Ordinary

Differential Equations Bapat: Linear Algebra and Linear Models (2nd ed.) Berberian: Fundamentals of Real Analysis BoltyanskiilEfremovich: Intuitive Combinatorial Topology. (Shenitzer, trans.) BoossIBleecker: Topology and Analysis Borkar: Probability Theory: An Advanced Course BottcherlSilbermann: Introduction to Large Truncated Toeplitz Matrices CarlesonlGamelin: Complex Dynamics Cecil: Lie Sphere Geometry: With Applications to Submanifolds Chae: Lebesgue Integration (2nd ed.) Charlap: Bieberbach Groups and Flat Manifolds Chern: Complex Manifolds Without Potential Theory Cohn: A Classical Invitation to Algebraic Numbers and Class Fields Curtis: Abstract Linear Algebra Curtis: Matrix Groups Debarre: Higher-Dimensional Algebraic Geometry Deitrnar: A First Course in Harmonic Analysis DiBenedetto: Degenerate Parabolic Equations Dimca: Singularities and Topology of Hypersurfaces Edwards: A Formal Background to Mathematics I alb Edwards: A Formal Background to Mathematics II alb Farenick: Algebras of Linear Transformations Foulds: Graph Theory Applications Friedman: Algebraic Surfaces and Holomorphic Vector Bundles Fnhnnann: A Polynomial Approach to Linear Algebra Gardiner: A First Course in Group Theory Garding/Tambour: Algebra for Computer Science Goldblatt: Orthogonality and Spacetime Geometry GustafsonlRao: Numerical Range: The Field of Values of Linear Operators

and Matrices Hahn: Quadratic Algebras, Clifford Algebras, and Arithmetic Witt Groups Heinonen: Lectures on Analysis on Metric Spaces Holmgren: A First Course in Discrete Dynamical Systems HoweITan: Non-Abelian Harmonic Analysis: Applications of SL(2, R) Howes: Modem Analysis and Topology Hsieh/Sibuya: Basic Theory of Ordinary Differential Equations HumilMiller: Second Course in Ordinary Differential Equations HurwitzlKritikos: Lectures on Number Theory Jennings: Modem Geometry with Applications Jones/Morris/Pearson: Abstract Algebra and Famous Impossibilities

(continued ajler index)

George Bachman Lawrence Narici Edward Beckenstein

Fourier and Wavelet Analysis

i Springer

George Bachman Professor Emeritus

of Mathematics Polytechnic University 5 Metrotech Center Brooklyn, NY 11201 USA

Lawrence Narici Department of Mathematics

and Computer Science St. John's University Jamaica, NY 11439 USA

Edward Beckenstein Science Division St. John's University Staten Island, NY 10301 USA

Editorial Board (North America):

S. Axler Mathematics Department San Francisco State University San Francisco, CA 94132 USA

K.A. Ribet Mathematics Department University of California at Berkeley Berkeley, CA 94720-3840 USA

F.W. Gehring Mathematics Department East Hall University of Michigan Ann Arbor, MI 48109-1109 USA

Mathematics Subject Classification (2000): 42Axx, 42Cxx, 41-xx

Library ofCongress Cataloging-in-Publication Data

Bachman, George, 1929-

Fourier and wavelet analysis / George Bachman, Lawrence Narici,

Edward Beckenstein.

p. cm. - (Universitext)

Includes bibliographical references and index.

ISBN 978-1-4612-6793-5 ISBN 978-1-4612-0505-0 (eBook)

DOI 10.1007/978-1-4612-0505-0

I. Fourier, analysis. 2. Wavelets (Mathematics) I. Beckenstein,

Edward, 1940--

11. Narici, Lawrence. III. Title. IV. Series.

QA403.5.B28 2000

515' .2433-dc21

99-36217

Printed on acid-free paper.

© 2000 Springer Science+Business Media New York Originally published by Springer-Verlag New York Berlin Heidelberg in 2000 Softcover reprint of the hardcover Ist edition 2000 All rights reserved. This work may not be translated or copied in whole or in part without the written permission of the publisher Springer Science+Business Media, LLC, 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 ifthe former are not especiaIly 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.

Production managed by Michael Koy; manufacturing supervised by Jeffrey Taub. Camera-ready copy prepared by the authors.

9 8 7 6 5 4 3 2 (Corrected second printing, 2002)

ISBN 978-1-4612-6793-5

Not long ago many thought that the mathematical world was created out of analytic functions. It was the Fourier series which disclosed a terra incognita in a second hemisphere. -E. B. van Vleck, 1914

The Fast Fourier transform-the most valuable numerical algorithm of our lifetime. -G. Strang, 1993

... wavelets are without any doubt an exciting and intuitive concept. This concept brings with it a new way of thinking.... -Yo Meyer, 1993

Foreword

Fourier, the nineteenth (and not the last!) child in his family, wanted to join an artillery regiment. His commoner status prevented it and he went on to other things. Goethe's dictum that boldness has a magic all its own found life in Fourier. He was so rash at times that his work was rejected by his peers (see the introduction to Chapter 4). He never worried about convergence and said that any periodic function could be expressed in a Fourier series. Nevertheless he was so original that others-Cauchy and Lagrange, among them-were inspired to attempt to place his creations on a firm foundation. They both failed. The first proof that Fourier series converged pointwise was Dirichlet's in 1829 for piecewise smooth functions (Sec. 4.6). As a result of Dirichlet's work, the idea of function was transmogrified. No more would it apply only to the aristocratic society of polynomials, exponentials and sines and cosines; disorderly conduct now had to be tolerated. By the mid-nineteenth century, it inspired (as a trigonometric series) Weierstrass's continuous-but-not-differentiable map (Sec. 4.3). It was such a shock at the time that Weierstrass was apparently in no hurry to disseminate it widely.

In order to generalize Dirichlet's pointwise convergence theorem for piecewise smooth functions to a wilder sort, Jordan invented the concept of 'function of bounded variation;' he proved his pointwise convergence theorem of Fourier series for such functions in 1881 (Sec. 4.6). As it became necessary to deal with this wider class of functions, the conception of integral was also transmuted. At Dirichlet's urging, it went from integral-asantiderivative to being defined as area under a curve. Cauchy developed the integral from this perspective for continuous functions. Riemann extended it to discontinuous functions, although not too discontinuous.

Fejer (1904) went beyond functions of bounded variation. He discovered

that for many functions I, f can be recovered by summing the arithmetic

means of its Fourier series; even if the Fourier series diverges at a point,

the series of arithmetic means converges to (J (t-) + I (t+» /2 (Sec. 4.15).

What happens at t's where the one-sided limits do not exist? By removing

the requirement concerning the existence of I (t-) and f (t+), Lebesgue

vi Foreword

globalized Fejer's theorem; he showed that the Fourier series for any f E Ld-7I", 7I"] converges (C, 1) to f (t) a.e. The desire to do this was part of the

reason that Lebesgue invented his integral; the theorem mentioned above was one of the first uses he made of it (Sec. 4.18). Denjoy, with the same motivation, extended the integral even further.

Concurrently, the emerging point of view that things could be decomposed into waves and then reconstituted infused not just mathematics but all of science. It is impossible to quantify the role that this perspective played in the development of the physics of the nineteenth and twentieth centuries, but it was certainly great. Imagine physics without it.

We develop the standard features of Fourier analysis-Fourier series, Fourier transform, Fourier sine and cosine transforms. We do NOT do it in the most elegant way. Instead, we develop it for the reader who has never seen them before. We cover more recent developments such as the discrete and fast Fourier transforms and wavelets in Chapters 6 and 7. Our treatment of these topics is strictly introductory, for the novice. (Wavelets for idiots?) To do them properly, especially the applications, would take at least a whole book.

What do you need to read the book? Not a lot of facts per se, but a little sophistication. We have helped ourselves to what we needed about the Lebesgue integral and given references. It's not much and if you don't know them exactly-if you know the analogous results (when they exist) for the Riemann integral-you can still read the book. We use some things about Hilbert space, too, and we have included a short development of what is needed in the first three short chapters. You can use them as a short course in functional analysis or start in Chapter 4 on Fourier series, referring to them on an as-needed basis. The chapters are sufficiently independent so that you could start in Chapter 5 (Fourier transform) or 6 (discrete, fast) or 7 (wavelets) and refer back as needed. One caveat: To appreciate Chapter 7, you should read the theory of the L2 Fourier transform in Chapter 5. The L2 theory is really quite pretty, anyway.

Notation: Our notation is all standard. On some rare occasions we use C for complement. If we say "by Exercise 3," we mean Exercise 3 at the end of the current section; otherwise we say "Exercise 2.4-3," meaning Exercise 3 at the end of Sec. 2.4. Hints are provided for lots (not all) of the exercises. Rather than a separate index of symbols, the symbols are blended into the index.

We prepared the book using Scientific Word and Scientific Workplace. The experience has been.. .interesting. We hope that the result is fun.

Contents

Foreword

v

1 Metric and Normed Spaces

1

1.1 Metric Spaces. . . .

1

1.2 Normed Spaces ...

9

1.3 Inner Product Spaces

12

1.4 Orthogonality ... .

18

1.5 Linear Isometry .. .

24

1.6 Holder and Minkowski Inequalities; Lp and Qp Spaces.

28

2 Analysis

35

2.1 Balls

35

2.2 Convergence and Continuity.

38

2.3 Bounded Sets . . . . . .

49

2.4 Closure and Closed Sets.

52

2.5 Open Sets . . . . . .

58

2.6 Completeness ... .

60

2.7 Uniform Continuity.

66

2.8 Compactness . . .

69

2.9 Equivalent Norms.

75

2.10 Direct Sums.

83

3 Bases

89

3.1 Best Approximation . . . . . . . . . . . . . . . . . .

90

3.2 Orthogonal Complements and the Projection Theorem.

94

3.3 Orthonormal Sequences

103

3.4 Orthonormal Bases . . . .

107

3.5 The Haar Basis ..... .

114

3.6 Unconditional Convergence

119

3.7 Orthogonal Direct Sums.

123

3.8 Continuous Linear Maps

126

3.9 Dual Spaces.

131

3.10 Adjoints . . . . . . . . .

135

viii Contents

4 Fourier Series

139

4.1 Warmup

143

4.2 Fourier Sine Series and Cosine Series

154

4.3 Smoothness .

159

4.4 The Riemann-Lebesgue Lemma.

169

4.5 The Dirichlet and Fourier Kernels.

174

4.6 Pointwise Convergence of Fourier Series.

188

4.7 Uniform Convergence.

202

4.8 The Gibbs Phenomenon.

207

4.9 A Divergent Fourier Series

210

4.10 Termwise Integration .

213

4.11 Trigonometric vs. Fourier Series.

218

4.12 Termwise Differentiation

221

4.13 Dido's Dilemma.

224

4.14 Other Kinds of Summability.

226

4.15 Fejer Theory

235

4.16 The Smoothing Effect of (C, 1) Summation

242

4.17 Weierstrass's Approximation Theorem.

244

4.18 Lebesgue's Pointwise Convergence Theorem.

245

4.19 Higher Dimensions

249

4.20 Convergence of Multiple Series

257

5 The Fourier Transform

263

5.1 The Finite Fourier Transform.

264

5.2 Convolution on T .

267

5.3 The Exponential Form of Lebesgue's Theorem.

273

5.4 Motivation and Definition.

275

5.5 Basics/Examples

278

5.6 The Fourier Transform and Residues

284

5.7 The Fourier Map

289

5.8 Convolution on R .

291

5.9 Inversion, Exponential Form

294

5.10 Inversion, Trigonometric Form

295

5.11 (C, 1) Summability for Integrals.

303

5.12 The Fejer-Lebesgue Inversion Theorem

306

5.13 Convergence Assistance.

317

5.14 Approximate Identity.

330

5.15 Transforms of Derivatives and Integrals.

334

5.16 Fourier Sine and Cosine Transforms.

340

5.17 Parseval's Identities.

351

5.18 The L2 Theory . . . . . . . . . . . 5.19 The Plancherel Theorem . . . . . . 5.20 Pointwise Inversion and Summability 5.21 A Sampling Theorem. 5.22 The Mellin Transform. 5.23 Variations. . . . . . .

6 The Discrete and Fast Fourier Transforms 6.1 The Discrete Fourier Transform . . . 6.2 The Inversion Theorem for the DFT. 6.3 Cyclic Convolution . . . . . . . . . 6.4 Fast Fourier Transform for N = 2k .. 6.5 The Fast Fourier Transform for N = RC.

7 Wavelets 7.1 Orthonormal Basis from One Function 7.2 Multiresolution Analysis. . . . . . . . 7.3 Mother Wavelets Yield Wavelet Bases. 7.4 From MRA to Mother Wavelet . . . . 7.5 Construction of a Scaling Function with Compact Support. . . 7.6 Shannon Wavelets. . . 7.7 Riesz Bases and MRAs 7.8 Franklin Wavelets. 7.9 Frames . . . . . . . . 7.10 Splines . . . . . . . . 7.11 The Continuous Wavelet Transform.

Index

Contents ix

356 361 366 372 375 378

383 383 390 396 399 406

411 413 414 419 422

435 448 449 459 464 476 480

497

Editorial Board (North America):

S. Axler F.W. Gehring

K.A. Ribet

Springer Science+Business Media, LLC

Universitext

Editors (North America): S. Axler, F.W. Gehring, and K.A. Ribet

Aksoy/Khamsi: Nonstandard Methods in Fixed Point Theory Andersson: Topics in Complex Analysis Aupetit: A Primer on Spectral Theory BaclunanlNaricilBeckenstein: Fourier and Wavelet Analysis Badescu: Algebraic Surfaces BalakrishnanlRanganathan: A Textbook of Graph Theory Balser: Formal Power Series and Linear Systems of Meromorphic Ordinary

Differential Equations Bapat: Linear Algebra and Linear Models (2nd ed.) Berberian: Fundamentals of Real Analysis BoltyanskiilEfremovich: Intuitive Combinatorial Topology. (Shenitzer, trans.) BoossIBleecker: Topology and Analysis Borkar: Probability Theory: An Advanced Course BottcherlSilbermann: Introduction to Large Truncated Toeplitz Matrices CarlesonlGamelin: Complex Dynamics Cecil: Lie Sphere Geometry: With Applications to Submanifolds Chae: Lebesgue Integration (2nd ed.) Charlap: Bieberbach Groups and Flat Manifolds Chern: Complex Manifolds Without Potential Theory Cohn: A Classical Invitation to Algebraic Numbers and Class Fields Curtis: Abstract Linear Algebra Curtis: Matrix Groups Debarre: Higher-Dimensional Algebraic Geometry Deitrnar: A First Course in Harmonic Analysis DiBenedetto: Degenerate Parabolic Equations Dimca: Singularities and Topology of Hypersurfaces Edwards: A Formal Background to Mathematics I alb Edwards: A Formal Background to Mathematics II alb Farenick: Algebras of Linear Transformations Foulds: Graph Theory Applications Friedman: Algebraic Surfaces and Holomorphic Vector Bundles Fnhnnann: A Polynomial Approach to Linear Algebra Gardiner: A First Course in Group Theory Garding/Tambour: Algebra for Computer Science Goldblatt: Orthogonality and Spacetime Geometry GustafsonlRao: Numerical Range: The Field of Values of Linear Operators

and Matrices Hahn: Quadratic Algebras, Clifford Algebras, and Arithmetic Witt Groups Heinonen: Lectures on Analysis on Metric Spaces Holmgren: A First Course in Discrete Dynamical Systems HoweITan: Non-Abelian Harmonic Analysis: Applications of SL(2, R) Howes: Modem Analysis and Topology Hsieh/Sibuya: Basic Theory of Ordinary Differential Equations HumilMiller: Second Course in Ordinary Differential Equations HurwitzlKritikos: Lectures on Number Theory Jennings: Modem Geometry with Applications Jones/Morris/Pearson: Abstract Algebra and Famous Impossibilities

(continued ajler index)

George Bachman Lawrence Narici Edward Beckenstein

Fourier and Wavelet Analysis

i Springer

George Bachman Professor Emeritus

of Mathematics Polytechnic University 5 Metrotech Center Brooklyn, NY 11201 USA

Lawrence Narici Department of Mathematics

and Computer Science St. John's University Jamaica, NY 11439 USA

Edward Beckenstein Science Division St. John's University Staten Island, NY 10301 USA

Editorial Board (North America):

S. Axler Mathematics Department San Francisco State University San Francisco, CA 94132 USA

K.A. Ribet Mathematics Department University of California at Berkeley Berkeley, CA 94720-3840 USA

F.W. Gehring Mathematics Department East Hall University of Michigan Ann Arbor, MI 48109-1109 USA

Mathematics Subject Classification (2000): 42Axx, 42Cxx, 41-xx

Library ofCongress Cataloging-in-Publication Data

Bachman, George, 1929-

Fourier and wavelet analysis / George Bachman, Lawrence Narici,

Edward Beckenstein.

p. cm. - (Universitext)

Includes bibliographical references and index.

ISBN 978-1-4612-6793-5 ISBN 978-1-4612-0505-0 (eBook)

DOI 10.1007/978-1-4612-0505-0

I. Fourier, analysis. 2. Wavelets (Mathematics) I. Beckenstein,

Edward, 1940--

11. Narici, Lawrence. III. Title. IV. Series.

QA403.5.B28 2000

515' .2433-dc21

99-36217

Printed on acid-free paper.

© 2000 Springer Science+Business Media New York Originally published by Springer-Verlag New York Berlin Heidelberg in 2000 Softcover reprint of the hardcover Ist edition 2000 All rights reserved. This work may not be translated or copied in whole or in part without the written permission of the publisher Springer Science+Business Media, LLC, 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 ifthe former are not especiaIly 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.

Production managed by Michael Koy; manufacturing supervised by Jeffrey Taub. Camera-ready copy prepared by the authors.

9 8 7 6 5 4 3 2 (Corrected second printing, 2002)

ISBN 978-1-4612-6793-5

Not long ago many thought that the mathematical world was created out of analytic functions. It was the Fourier series which disclosed a terra incognita in a second hemisphere. -E. B. van Vleck, 1914

The Fast Fourier transform-the most valuable numerical algorithm of our lifetime. -G. Strang, 1993

... wavelets are without any doubt an exciting and intuitive concept. This concept brings with it a new way of thinking.... -Yo Meyer, 1993

Foreword

Fourier, the nineteenth (and not the last!) child in his family, wanted to join an artillery regiment. His commoner status prevented it and he went on to other things. Goethe's dictum that boldness has a magic all its own found life in Fourier. He was so rash at times that his work was rejected by his peers (see the introduction to Chapter 4). He never worried about convergence and said that any periodic function could be expressed in a Fourier series. Nevertheless he was so original that others-Cauchy and Lagrange, among them-were inspired to attempt to place his creations on a firm foundation. They both failed. The first proof that Fourier series converged pointwise was Dirichlet's in 1829 for piecewise smooth functions (Sec. 4.6). As a result of Dirichlet's work, the idea of function was transmogrified. No more would it apply only to the aristocratic society of polynomials, exponentials and sines and cosines; disorderly conduct now had to be tolerated. By the mid-nineteenth century, it inspired (as a trigonometric series) Weierstrass's continuous-but-not-differentiable map (Sec. 4.3). It was such a shock at the time that Weierstrass was apparently in no hurry to disseminate it widely.

In order to generalize Dirichlet's pointwise convergence theorem for piecewise smooth functions to a wilder sort, Jordan invented the concept of 'function of bounded variation;' he proved his pointwise convergence theorem of Fourier series for such functions in 1881 (Sec. 4.6). As it became necessary to deal with this wider class of functions, the conception of integral was also transmuted. At Dirichlet's urging, it went from integral-asantiderivative to being defined as area under a curve. Cauchy developed the integral from this perspective for continuous functions. Riemann extended it to discontinuous functions, although not too discontinuous.

Fejer (1904) went beyond functions of bounded variation. He discovered

that for many functions I, f can be recovered by summing the arithmetic

means of its Fourier series; even if the Fourier series diverges at a point,

the series of arithmetic means converges to (J (t-) + I (t+» /2 (Sec. 4.15).

What happens at t's where the one-sided limits do not exist? By removing

the requirement concerning the existence of I (t-) and f (t+), Lebesgue

vi Foreword

globalized Fejer's theorem; he showed that the Fourier series for any f E Ld-7I", 7I"] converges (C, 1) to f (t) a.e. The desire to do this was part of the

reason that Lebesgue invented his integral; the theorem mentioned above was one of the first uses he made of it (Sec. 4.18). Denjoy, with the same motivation, extended the integral even further.

Concurrently, the emerging point of view that things could be decomposed into waves and then reconstituted infused not just mathematics but all of science. It is impossible to quantify the role that this perspective played in the development of the physics of the nineteenth and twentieth centuries, but it was certainly great. Imagine physics without it.

We develop the standard features of Fourier analysis-Fourier series, Fourier transform, Fourier sine and cosine transforms. We do NOT do it in the most elegant way. Instead, we develop it for the reader who has never seen them before. We cover more recent developments such as the discrete and fast Fourier transforms and wavelets in Chapters 6 and 7. Our treatment of these topics is strictly introductory, for the novice. (Wavelets for idiots?) To do them properly, especially the applications, would take at least a whole book.

What do you need to read the book? Not a lot of facts per se, but a little sophistication. We have helped ourselves to what we needed about the Lebesgue integral and given references. It's not much and if you don't know them exactly-if you know the analogous results (when they exist) for the Riemann integral-you can still read the book. We use some things about Hilbert space, too, and we have included a short development of what is needed in the first three short chapters. You can use them as a short course in functional analysis or start in Chapter 4 on Fourier series, referring to them on an as-needed basis. The chapters are sufficiently independent so that you could start in Chapter 5 (Fourier transform) or 6 (discrete, fast) or 7 (wavelets) and refer back as needed. One caveat: To appreciate Chapter 7, you should read the theory of the L2 Fourier transform in Chapter 5. The L2 theory is really quite pretty, anyway.

Notation: Our notation is all standard. On some rare occasions we use C for complement. If we say "by Exercise 3," we mean Exercise 3 at the end of the current section; otherwise we say "Exercise 2.4-3," meaning Exercise 3 at the end of Sec. 2.4. Hints are provided for lots (not all) of the exercises. Rather than a separate index of symbols, the symbols are blended into the index.

We prepared the book using Scientific Word and Scientific Workplace. The experience has been.. .interesting. We hope that the result is fun.

Contents

Foreword

v

1 Metric and Normed Spaces

1

1.1 Metric Spaces. . . .

1

1.2 Normed Spaces ...

9

1.3 Inner Product Spaces

12

1.4 Orthogonality ... .

18

1.5 Linear Isometry .. .

24

1.6 Holder and Minkowski Inequalities; Lp and Qp Spaces.

28

2 Analysis

35

2.1 Balls

35

2.2 Convergence and Continuity.

38

2.3 Bounded Sets . . . . . .

49

2.4 Closure and Closed Sets.

52

2.5 Open Sets . . . . . .

58

2.6 Completeness ... .

60

2.7 Uniform Continuity.

66

2.8 Compactness . . .

69

2.9 Equivalent Norms.

75

2.10 Direct Sums.

83

3 Bases

89

3.1 Best Approximation . . . . . . . . . . . . . . . . . .

90

3.2 Orthogonal Complements and the Projection Theorem.

94

3.3 Orthonormal Sequences

103

3.4 Orthonormal Bases . . . .

107

3.5 The Haar Basis ..... .

114

3.6 Unconditional Convergence

119

3.7 Orthogonal Direct Sums.

123

3.8 Continuous Linear Maps

126

3.9 Dual Spaces.

131

3.10 Adjoints . . . . . . . . .

135

viii Contents

4 Fourier Series

139

4.1 Warmup

143

4.2 Fourier Sine Series and Cosine Series

154

4.3 Smoothness .

159

4.4 The Riemann-Lebesgue Lemma.

169

4.5 The Dirichlet and Fourier Kernels.

174

4.6 Pointwise Convergence of Fourier Series.

188

4.7 Uniform Convergence.

202

4.8 The Gibbs Phenomenon.

207

4.9 A Divergent Fourier Series

210

4.10 Termwise Integration .

213

4.11 Trigonometric vs. Fourier Series.

218

4.12 Termwise Differentiation

221

4.13 Dido's Dilemma.

224

4.14 Other Kinds of Summability.

226

4.15 Fejer Theory

235

4.16 The Smoothing Effect of (C, 1) Summation

242

4.17 Weierstrass's Approximation Theorem.

244

4.18 Lebesgue's Pointwise Convergence Theorem.

245

4.19 Higher Dimensions

249

4.20 Convergence of Multiple Series

257

5 The Fourier Transform

263

5.1 The Finite Fourier Transform.

264

5.2 Convolution on T .

267

5.3 The Exponential Form of Lebesgue's Theorem.

273

5.4 Motivation and Definition.

275

5.5 Basics/Examples

278

5.6 The Fourier Transform and Residues

284

5.7 The Fourier Map

289

5.8 Convolution on R .

291

5.9 Inversion, Exponential Form

294

5.10 Inversion, Trigonometric Form

295

5.11 (C, 1) Summability for Integrals.

303

5.12 The Fejer-Lebesgue Inversion Theorem

306

5.13 Convergence Assistance.

317

5.14 Approximate Identity.

330

5.15 Transforms of Derivatives and Integrals.

334

5.16 Fourier Sine and Cosine Transforms.

340

5.17 Parseval's Identities.

351

5.18 The L2 Theory . . . . . . . . . . . 5.19 The Plancherel Theorem . . . . . . 5.20 Pointwise Inversion and Summability 5.21 A Sampling Theorem. 5.22 The Mellin Transform. 5.23 Variations. . . . . . .

6 The Discrete and Fast Fourier Transforms 6.1 The Discrete Fourier Transform . . . 6.2 The Inversion Theorem for the DFT. 6.3 Cyclic Convolution . . . . . . . . . 6.4 Fast Fourier Transform for N = 2k .. 6.5 The Fast Fourier Transform for N = RC.

7 Wavelets 7.1 Orthonormal Basis from One Function 7.2 Multiresolution Analysis. . . . . . . . 7.3 Mother Wavelets Yield Wavelet Bases. 7.4 From MRA to Mother Wavelet . . . . 7.5 Construction of a Scaling Function with Compact Support. . . 7.6 Shannon Wavelets. . . 7.7 Riesz Bases and MRAs 7.8 Franklin Wavelets. 7.9 Frames . . . . . . . . 7.10 Splines . . . . . . . . 7.11 The Continuous Wavelet Transform.

Index

Contents ix

356 361 366 372 375 378

383 383 390 396 399 406

411 413 414 419 422

435 448 449 459 464 476 480

497