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A First Course in Fourier Analysis

This unique book provides a meaningful resource for applied mathematics through Fourier analysis. It develops a unified theory of discrete and continuous (univariate) Fourier analysis, the fast Fourier transform, and a powerful elementary theory of generalized functions, including the use of weak limits. It then shows how these mathematical ideas can be used to expedite the study of sampling theory, PDEs, wavelets, probability, diffraction, etc. Unique features include a unified development of Fourier synthesis/analysis for functions on R, Tp , Z, and PN ; an unusually complete development of the Fourier transform calculus (for finding Fourier transforms, Fourier series, and DFTs); memorable derivations of the FFT; a balanced treatment of generalized functions that fosters mathematical understanding as well as practical working skills; a careful introduction to Shannon’s sampling theorem and modern variations; a study of the wave equation, diffusion equation, and diffraction equation by using the Fourier transform calculus, generalized functions, and weak limits; an exceptionally efficient development of Daubechies’ compactly supported orthogonal wavelets; generalized probability density functions with corresponding versions of Bochner’s theorem and the central limit theorem; and a real-world application of Fourier analysis to the study of musical tones. A valuable reference on Fourier analysis for a variety of scientific professionals, including Mathematicians, Physicists, Chemists, Geologists, Electrical Engineers, Mechanical Engineers, and others. David Kammler is a Professor and Distinguished Teacher in the Mathematics Department at Southern Illinois University.

A First Course in Fourier Analysis David W. Kammler Department of Mathematics Southern Illinois University at Carbondale

CAMBRIDGE UNIVERSITY PRESS

Cambridge, New York, Melbourne, Madrid, Cape Town, Singapore, São Paulo Cambridge University Press The Edinburgh Building, Cambridge CB2 8RU, UK Published in the United States of America by Cambridge University Press, New York www.cambridge.org Information on this title: www.cambridge.org/9780521883405 © D. W. Kammler 2007 This publication is in copyright. Subject to statutory exception and to the provision of relevant collective licensing agreements, no reproduction of any part may take place without the written permission of Cambridge University Press. First published in print format 2008

ISBN-13 978-0-511-37689-4

eBook (EBL)

ISBN-13

978-0-521-88340-5

hardback

ISBN-13

978-0-521-70979-8

paperback

Cambridge University Press has no responsibility for the persistence or accuracy of urls for external or third-party internet websites referred to in this publication, and does not guarantee that any content on such websites is, or will remain, accurate or appropriate.

Mathematics: Source and Substance Profound study of nature is the most fertile source of mathematical discoveries. Joseph Fourier, The Analytical Study of Heat, p. 7

Mathematics is the science of patterns. The mathematician seeks patterns in number, in space, in science, in computers, and in imagination. Mathematical theories explain the relations among patterns; functions and maps, operators and morphisms bind one type of pattern to another to yield lasting mathematical structures. Applications of mathematics use these patterns to explain and predict natural phenomena that fit the patterns. Patterns suggest other patterns, often yielding patterns of patterns. In this way mathematics follows its own logic, beginning with patterns from science and completing the portrait by adding all patterns that derive from initial ones. Lynn A. Steen, The science of patterns, Science 240(1988), 616.

Contents Preface

xi

The Mathematical Core Chapter 1 1.1 1.2 1.3 1.4 1.5

Chapter 2 2.1 2.2 2.3 2.4

Chapter 3 3.1 3.2 3.3

Fourier’s representation for functions on R, Tp , Z, and PN

1

Synthesis and analysis equations Examples of Fourier’s representation The Parseval identities and related results The Fourier–Poisson cube The validity of Fourier’s representation Further reading Exercises

1 12 23 31 37 59 61

Convolution of functions on R, Tp , Z, and PN

89

Formal definitions of f ∗ g, f
g Computation of f ∗ g Mathematical properties of the convolution product Examples of convolution and correlation Further reading Exercises

89 91 102 107 115 116

The calculus for finding Fourier transforms of functions on R Using the definition to find Fourier transforms Rules for finding Fourier transforms Selected applications of the Fourier transform calculus Further reading Exercises

vii

129 129 134 147 155 156

viii

Contents

Chapter 4 4.1 4.2 4.3 4.4

Chapter 5 5.1 5.2 5.3 5.4 5.5 5.6

Chapter 6 6.1 6.2 6.3 6.4 6.5 6.6 6.7

Chapter 7 7.1 7.2 7.3 7.4 7.5 7.6 7.7 7.8

The calculus for finding Fourier transforms of functions on Tp , Z, and PN

173

Fourier series Selected applications of Fourier series Discrete Fourier transforms Selected applications of the DFT calculus Further reading Exercises

173 190 196 212 216 217

Operator identities associated with Fourier analysis

239

The concept of an operator identity Operators generated by powers of F Operators related to complex conjugation Fourier transforms of operators Rules for Hartley transforms Hilbert transforms Further reading Exercises

239 243 251 255 263 266 271 272

The fast Fourier transform Pre-FFT computation of the DFT Derivation of the FFT via DFT rules The bit reversal permutation Sparse matrix factorization of F when N = 2m Sparse matrix factorization of H when N = 2m Sparse matrix factorization of F when N = P1 P2 · · · Pm Kronecker product factorization of F Further reading Exercises

Generalized functions on R The concept of a generalized function Common generalized functions Manipulation of generalized functions Derivatives and simple differential equations The Fourier transform calculus for generalized functions Limits of generalized functions Periodic generalized functions Alternative definitions for generalized functions Further reading Exercises

291 291 296 303 310 323 327 338 345 345

367 367 379 389 405 413 427 440 450 452 453