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Harmonic Analysis from Fourier to Wavelets STUDENT MATHEMATICAL LIBRARY ⍀ IAS/PARK CITY MATHEMATICAL SUBSERIES Volume 63 Harmonic Analysis From Fourier to Wavelets María Cristina Pereyra Lesley A. Ward American Mathematical Society Institute for Advanced Study Harmonic Analysis From Fourier to Wavelets STUDENT MATHEMATICAL LIBRARY IAS/PARK CITY MATHEMATICAL SUBSERIES Volume 63 Harmonic Analysis From Fourier to Wavelets María Cristina Pereyra Lesley A. Ward American Mathematical Society, Providence, Rhode Island Institute for Advanced Study, Princeton, New Jersey Editorial Board of the Student Mathematical Library Gerald B. Folland Brad G. Osgood (Chair) Robin Forman John Stillwell Series Editor for the Park City Mathematics Institute John Polking 2010 Mathematics Subject Classification. Primary 42–01; Secondary 42–02, 42Axx, 42B25, 42C40. The anteater on the dedication page is by Miguel. The dragon at the back of the book is by Alexander. For additional information and updates on this book, visit www.ams.org/bookpages/stml-63 Library of Congress Cataloging-in-Publication Data Pereyra, Mar´ıa Cristina. Harmonic analysis : from Fourier to wavelets / Mar´ıa Cristina Pereyra, Lesley A. Ward. p. cm. — (Student mathematical library ; 63. IAS/Park City mathematical subseries) Includes bibliographical references and indexes. ISBN 978-0-8218-7566-7 (alk. paper) 1. Harmonic analysis—Textbooks. I. Ward, Lesley A., 1963– II. Title. QA403.P44 2012 515.2433—dc23 2012001283 Copying and reprinting. Individual readers of this publication, and nonprofit libraries acting for them, are permitted to make fair use of the material, such as to copy a chapter for use in teaching or research. Permission is granted to quote brief passages from this publication in reviews, provided the customary acknowledgment of the source is given. Republication, systematic copying, or multiple reproduction of any material in this publication is permitted only under license from the American Mathematical Society. Requests for such permission should be addressed to the Acquisitions Department, American Mathematical Society, 201 Charles Street, Providence, Rhode Island 02904- 2294 USA. Requests can also be made by e-mail to [email protected]. c 2012 by the American Mathematical Society. All rights reserved. The American Mathematical Society retains all rights except those granted to the United States Government. Printed in the United States of America. ∞ The paper used in this book is acid-free and falls within the guidelines established to ensure permanence and durability. Visit the AMS home page at http://www.ams.org/ 10987654321 171615141312 To Tim, Nicolás, and Miguel To Jorge and Alexander Contents List of figures xi List of tables xiii IAS/Park City Mathematics Institute xv Preface xvii Suggestions for instructors xx Acknowledgements xxii Chapter 1. Fourier series: Some motivation 1 §1.1. An example: Amanda calls her mother 2 §1.2. The main questions 7 §1.3. Fourier series and Fourier coefficients 10 §1.4. History, and motivation from the physical world 15 §1.5. Project: Other physical models 19 Chapter 2. Interlude: Analysis concepts 21 §2.1. Nested classes of functions on bounded intervals 22 §2.2. Modes of convergence 38 §2.3. Interchanging limit operations 45 §2.4. Density 49 §2.5. Project: Monsters, Take I 52 vii viii Contents Chapter 3. Pointwise convergence of Fourier series 55 §3.1. Pointwise convergence: Why do we care? 55 §3.2. Smoothness vs. convergence 59 §3.3. A suite of convergence theorems 68 §3.4. Project: The Gibbs phenomenon 73 §3.5. Project: Monsters, Take II 74 Chapter 4. Summability methods 77 §4.1. Partial Fourier sums and the Dirichlet kernel 78 §4.2. Convolution 82 §4.3. Good kernels, or approximations of the identity 90 §4.4. Fejér kernels and Cesàro means 96 §4.5. Poisson kernels and Abel means 99 §4.6. Excursion into Lp(T) 101 §4.7. Project: Weyl’s Equidistribution Theorem 102 §4.8. Project: Averaging and summability methods 104 Chapter 5. Mean-square convergence of Fourier series 107 §5.1. Basic Fourier theorems in L2(T) 108 §5.2. Geometry of the Hilbert space L2(T) 110 §5.3. Completeness of the trigonometric system 115 §5.4. Equivalent conditions for completeness 122 §5.5. Project: The isoperimetric problem 126 Chapter 6. A tour of discrete Fourier and Haar analysis 127 §6.1. Fourier series vs. discrete Fourier basis 128 §6.2. Short digression on dual bases in CN 134 §6.3. The Discrete Fourier Transform and its inverse 137 §6.4. The Fast Fourier Transform (FFT) 138 §6.5. The discrete Haar basis 146 §6.6. The Discrete Haar Transform 151 §6.7. The Fast Haar Transform 152 Contents ix §6.8. Project: Two discrete Hilbert transforms 157 §6.9. Project: Fourier analysis on finite groups 159 Chapter 7. The Fourier transform in paradise 161 §7.1. From Fourier series to Fourier integrals 162 §7.2. The Schwartz class 164 §7.3. The time–frequency dictionary for S(R) 167 §7.4. The Schwartz class and the Fourier transform 172 §7.5. Convolution and approximations of the identity 175 §7.6. The Fourier Inversion Formula and Plancherel 179 §7.7. Lp norms on S(R) 184 §7.8. Project: A bowl of kernels 187 Chapter 8. Beyond paradise 189 §8.1. Continuous functions of moderate decrease 190 §8.2. Tempered distributions 193 §8.3. The time–frequency dictionary for S(R) 197 §8.4. The delta distribution 202 §8.5. Three applications of the Fourier transform 205 §8.6. Lp(R) as distributions 213 §8.7. Project: Principal value distribution 1/x 217 §8.8. Project: Uncertainty and primes 218 Chapter 9. From Fourier to wavelets, emphasizing Haar 221 §9.1. Strang’s symphony analogy 222 §9.2. The windowed Fourier and Gabor bases 224 §9.3. The wavelet transform 230 §9.4. Haar analysis 236 §9.5. Haar vs. Fourier 250 §9.6. Project: Local cosine and sine bases 257 §9.7. Project: Devil’s advocate 257 §9.8. Project: Khinchine’s Inequality 258 x Contents Chapter 10. Zooming properties of wavelets 261 §10.1. Multiresolution analyses (MRAs) 262 §10.2. Two applications of wavelets, one case study 271 §10.3. From MRA to wavelets: Mallat’s Theorem 278 §10.4. How to find suitable MRAs 288 §10.5. Projects: Twin dragon; infinite mask 298 §10.6. Project: Linear and nonlinear approximations 299 Chapter 11. Calculating with wavelets 303 §11.1. The Haar multiresolution analysis 303 §11.2. The cascade algorithm 308 §11.3. Filter banks, Fast Wavelet Transform 314 §11.4. A wavelet library 323 §11.5. Project: Wavelets in action 328 Chapter 12. The Hilbert transform 329 §12.1. In the frequency domain: A Fourier multiplier 330 §12.2. In the time domain: A singular integral 333 §12.3. In the Haar domain: An average of Haar shifts 336 §12.4. Boundedness on Lp of the Hilbert transform 341 §12.5. Weak boundedness on L1(R) 346 §12.6. Interpolation and a festival of inequalities 352 §12.7. Some history to conclude our journey 358 §12.8. Project: Edge detection and spectroscopy 365 §12.9. Projects: Harmonic analysis for researchers 366 Appendix. Useful tools 371 §A.1. Vector spaces, norms, inner products 371 §A.2. Banach spaces and Hilbert spaces 377 §A.3. Lp(R), density, interchanging limits on R 382 Bibliography 391 Name index 401 Subject index 403 List of figures 1.1 A toy voice signal 3 1.2 Plot of a voice recording 6 1.3 Graph of a periodic ramp function 13 1.4 Sketch of a one-dimensional bar 17 2.1 Graph of a step function 23 2.2 Approximating a function by step functions 25 2.3 Ladder of nested classes of functions f : T → C 33 2.4 Relations between five modes of convergence 42 2.5 Pointwise but not uniform convergence 43 2.6 Pointwise but not mean, L2, or uniform convergence 43 2.7 Convergence in mean but not pointwise 44 2.8 A continuous function approximating a step function 50 3.1 Partial Fourier sums SN f for the ramp function 57 3.2 Partial Fourier sums for the plucked string function 68 4.1 Graphs of Dirichlet kernels DN for N =1,3,8 80 4.2 Dirichlet kernels DN grow logarithmically in N 81 4.3 Convolution with a kernel 93 xi xii List of figures 4.4 Graphs of rectangular kernels Kn for n =2,4,8 94 4.5 Graphs of Fejér kernels FN for N =1,3,5 97 4.6 Graphs of Poisson kernels Pr for r =1/2, 2/3, 4/5 99 5.1 Projection of a vector onto a subspace 125 7.1 Graphs of Gauss kernels Gt for t =1, 3, 7.5 179 2 9.1 A Morlet wavelet given by ψ(x)=e−t /2 cos(5t) 221 9.2 The imaginary part of a Gabor function 227 9.3 The Haar wavelet h(x) 232 9.4 Daubechies wavelet function ψ, for the db2 wavelet 234 9.5 Parent interval I and its children Il and Ir 237 9.6 Graphs of two Haar functions 238 9.7 Graphs of f, P−1f, P0f,andQ0f 241 9.8 Nested dyadic intervals containing a given point 244 10.1 Daubechies functions for filter lengths 4, 8, 12 267 10.2 A fingerprint and a magnified detail 276 10.3 JPEG compression and wavelet compression 277 11.1 A wavelet decomposition of a subspace 306 11.2 The Haar scaling function and the hat function 309 11.3 Three steps of a cascade with a bad filter 311 11.4 Two steps of a cascade with the Haar filter 313 11.5 A cascade where half-boxes lead to a hat 314 12.1 The Hilbert transform of a characteristic function 349 12.2 Region of integration for Fubini’s Theorem 351 12.3 The Hausdorff–Young Inequality via interpolation 354 12.4 A generalized Hölder Inequality via interpolation 354 12.5 The Hilbert transform via the Poisson kernel 360 List of tables 1.1 Physical models 20 4.1 The time–frequency dictionary for Fourier series 89 7.1 The time–frequency dictionary in S(R) 169 8.1 New tempered distributions from old 199 8.2 The time–frequency dictionary in S(R) 201 8.3 Effect of the Fourier transform on Lp spaces 215 xiii IAS/Park City Mathematics Institute The IAS/Park City Mathematics Institute (PCMI) was founded in 1991 as part of the “Regional Geometry Institute” initiative of the National Science Foundation.
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