Fourier analysis and its applications, s axler, f w gehring, k a ribet

Graduate Texts in Mathematics 223 Editorial Board S Axler F.W Gehring K.A Ribet Graduate Texts in Mathematics 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 TAKEUTJIZARING Introduction to Axiomatic Set Theory 2nd ed 0XTOBY Measure and Category 2nd ed ScHAEFER Topological Vector Spaces 2nd ed Hn.roNISTAMMBACH A Course in Homological Algebra 2nd ed MAc LANE Categories for the Working Mathematician 2nd ed Humms/PIPER Projective Planes J.-P SERRE A Course in Arithmetic TAKEUTJIZARING Axiomatic Set Theory HUMPHREYS Jntroduction to Lie Algebras and Representation Theory CoHEN A Course in Simple Homotopy Theory CoNWAY Functions ofOne Complex Variable 2nd ed BEALS Advanced Mathematical Analysis ANDERSON!Fuu.ER Rings and Categories of Modules 2nd ed GoLUBITSKYIGUILLEMIN Stable Mappings and Their Singularities BERBERIAN Lectures in Functional Analysis and Operator Theory WINTER The Structure ofFields RoSENBLATT Random Processes 2nd ed HAI.Mos Measure Theory HAI.Mos A Hilbert Space Prob1em Book 2nded HUSEMOLLER Fibre Bundles 3rd ed HUMPHREYS Linear Algebraic Groups BARNES!MACK An Algebraic Introduction to Mathematica1 Logic GREUB Linear Algebra 4th ed HoLMES Geometric Functional Analysis and lts Applications HEwm/STROMBERG Real and Abstract Analysis MANES Algebraic Theories KELLEY General Topology :lARISKIISAMUEL Commutative Algebra Vol.l ZAR.IsKIISAMUEL Commutative Algebra Voi II JACOBSON Lectures in Abstract Algebra Basic Concepts JACOBSON Lectures in Abstract Algebra D Linear Algebra JACOBSON Lectures in Abstract Algebra III Theory ofFields and Galois Theory HlRsCH Differential Topology 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 SPITZER Principles ofRandom Walk 2nded Al.BxANDERIWBRMBR Several Complex Variables and Banach Algebras 3rd ed Km.u!YINAMIOKA et al Linear Topologica) Spaces MoNK Mathematical Logic GRAUERTIFRITzsCHE Several Complex Variables AilVESON An Jnvitation to C*-Aigebras KEMENY/SNEU.IKNAPP Denumerable Markov Chains 2nd ed APosTOL Modular Functions and Dirichlet Series in Number Theory 2nded 1.-P SERRE Linear Representations of Finite Groups GILLMANIJERISON Rings ofContinuous Functions KENDIG Elementary Algebraic Geometry LotM! 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Probability Theory n 4th ed MoiSE Geometric Topology in Dimensions and SACHSIWu General Relativity for Mathemaficians GRUENBERGIWEIR Linear Geometry 2nded EDWARDS Fermat's Last Theorem KLINGENBERG A Course in Differential Geometry 52 HARTSHORNE Algebraic Geometry 53 MANIN A Course in Mathematical Logic 54 GRAVERIWATKINS Combinatorics with Emphasis on the Theory of Graphs 55 BROwN!PEARCY lntroduction to Operator Theory 1: Elements of Functional Analysis 56 MASSEY Algebraic Topology: An Introduction 57 CRoWELLIFox Introduction to Knot Theory 58 Kosmz p-adic Numbers, p-adic Analysis, and Zeta-Functions 2nd ed 59 I ANG Cyclotomic Fields 60 ARNOLD Mathematical Methods in Classical Mechanics 2nd ed 61 WHITEHEAD Elements of Homotopy Theory 62 KARGAPOLOV/MERLZJAKOV Fundamentals of the Theory of Groups 63 BOLLOBAS Graph Theory (continued after index) Anders Vretblad Fourier Analysis and lts Applications ~Springer Anders Vretblad Department of Mathematics Uppsala University Box 480 SE-751 06 Uppsala Sweden anders.vretblad@math.uu.se Editorial Board: S Axler Mathematics Department San Francisco State University San Francisco, CA 94132 USA axler@ sfsu.edu F.W Gehring Mathematics Department East Hall University of Michigan Ann Arbor, MI 48109 USA fgehring@math.lsa.umich.edu K.A Ribet Mathematics Department University of California, Berkeley Berkeley, CA 94720-3840 USA ribet@math.berkeley edu Mathematics Subject Classification (2000): 42-01 Library of Congress Cataloging-in-Publication Data Vretblad, Anders Fourier analysis and its applications Anders Vretblad p cm Includes bibliographical references and index ISBN 0-387-00836-5 (hc : alk paper) Fourier analysis I Title QA403.5 V74 2003 515'2433-dc21 2003044941 ISBN 0-387-00836-5 Printed on acid-free paper © 2003 Springer-Verlag New York, Inc Ali 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, Inc., 233 Spring Street, New York, NY 10013, 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 in this publication of trade names, trademarks, service marks, and similar terms, even if they are not identified as such, is not to be taken as an expression of opinion as to whether or not they are subject to proprietary rights 98765432 springeronline.com Corrected second printing, 2005 SPIN 10920442 To YNGVE DOMAR, my teacher, mentor, and friend Preface The classical theory of Fourier series and integrals, as well as Laplace transforms, is of great importance for physical and technical applications, and its mathematical beauty makes it an interesting study for pure mathematicians as well I have taught courses on these subjects for decades to civil engineering students, and also mathematics majors, and the present volume can be regarded as my collected experiences from this work There is, of course, an unsurpassable book on Fourier analysis, the treatise by Katznelson from 1970 That book is, however, aimed at mathematically very mature students and can hardly be used in engineering courses On the other end of the scale, there are a number of more-or-less cookbookstyled books, where the emphasis is almost entirely on applications I have felt the need for an alternative in between these extremes: a text for the ambitious and interested student, who on the other hand does not aspire to become an expert in the field There exist a few texts that fulfill these requirements (see the literature list at the end of the book), but they not include all the topics I like to cover in my courses, such as Laplace transforms and the simplest facts about distributions The reader is assumed to have studied real calculus and linear algebra and to be familiar with complex numbers and uniform convergence On the other hand, we not require the Lebesgue integral Of course, this somewhat restricts the scope of some of the results proved in the text, but the reader who does master Lebesgue integrals can probably extrapolate the theorems Our ambition has been to prove as much as possible within these restrictions viii Some knowledge of the simplest distributions, such as point masses and dipoles, is essential for applications I have chosen to approach this matter in two separate ways: first, in an intuitive way that may be sufficient for engineering students, in star-marked sections of Chapter and subsequent chapters; secondly, in a more strict way, in Chapter 8, where at least the fundaments are given in a mathematically correct way Only the one-dimensional case is treated This is not intended to be more than the merest introduction, to whet the reader's appetite Acknowledgements In my work I have, of course, been inspired by existing literature In particular, I want to mention a book by Arne Broman, lntroduction to Partial Differential Equations (Addison-Wesley, 1970), a compendium by Jan Petersson of the Chalmers Institute of Technology in Gothenburg, and also a compendium from the Royal Institute of Technology in Stockholm, by Jockum Aniansson, Michael Benedicks, and Karim Daho I am grateful to my colleagues and friends in Uppsala First of all Professor Yngve Domar, who has been my teacher and mentor, and who introduced me to the field The book is dedicateq to him am also particularly indebted to Gunnar Berg, Christer O Kiselman, Anders Kăllstrom, Lars-Ăke Lindahl, and Lennart Salling Bengt Carlsson has helped with ideas for the applications to control theory The problems have been worked and re-worked by Jonas Bjermo and Daniel Domert If any incorrect answers still remain, the blame is mine Finally, special thanks go to three former students at Uppsala University, Mikael Nilsson, Matthias Palmer, and Magnus Sandberg They used an early version of the text and presented me with very constructive criticism This actually prompted me to pursue my work on the text, and to translate it into English Uppsala, Sweden January 2003 Anders Vret blad Contents vii Preface Introduction 101 The classical partial differential equations 1.2 Well-posed problems 1.3 The one-dimensional wave equation 1.4 Fourier's method o o o o o o o o 1 o o o o o o o o o o o o o Preparations 201 Complex exponentials 202 Complex-valued functions of a real variable 203 Cesaro summation of series 204 Positive summation kernels 205 The Riemann-Lebesgue lemma 206 *Some simple distributions 20 *Computing with o o o o o o o o o o o o o o o Laplace and Z transforms 301 The Laplace transform 302 Operations 303 Applications to differential equations 3.4 Convolution 305 *Laplace transforms of distributions 306 The Z transform o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o 15 o 15 17 20 22 25 27 32 39 39 42 47 53 57 60 x Contents Applications in control theory Summary of Chapter Fourier series 4.1 Definitions 4.2 Dirichlet's and Fejer's kernels; uniqueness 4.3 Differentiable functions 4.4 Pointwise convergence 4.5 Formulae for other periods 4.6 Some worked examples The Gibbs phenomenon 4.8 *Fourier series for distributions Summary of Chapter 67 70 73 73 80 84 86 90 91 93 96 100 L Theory 5.1 Linear spaces over the complex numbers 5.2 Orthogonal projections 5.3 Some examples 5.4 The Fourier system is complete 5.5 Legendre polynomials 5.6 Other classical orthogonal polynomials Summary of Chapter 105 Separation of variables 6.1 The solution of Fourier's problem 6.2 Variations on Fourier's theme 6.3 The Dirichlet problem in the unit disk 6.4 Sturm-Liouville problems 6.5 Some singular Sturm-Liouville problems Summary of Chapter 137 Fourier transforms 7.1 Introduction 7.2 Definition of the Fourier transform 7.3 Properties 7.4 The inversion theorem 7.5 The convolution theorem 7.6 Plancherel's formula 7.7 Application 7.8 Application 7.9 Application 3: The sampling theorem 7.10 *Connection with the Laplace transform 7.11 *Distributions and Fourier transforms Summary of Chapter 165 105 110 114 119 123 127 130 137 139 148 153 159 160 165 166 168 171 176 180 182 185 187 188 190 192 258 Appendix D Answers to selected exercises 2.40 f"'(x) = 24x(H(x + 1)- H(x -1) + 8(8(x + 1)- o(x -1)) Chapter 3.3 i(s) = (1 + e- rs)/(s + 1) 34 ( ) · · a s3 - + 2e-• 3.6 f(s) = - s 3.7 J(s) = ~ ln1 : (b) s + 4s: + 24 s (c) ~ _ s s2 + 1re-1rs 2s(e-1rs + 1) 3.9 f(s) = s2 + + (s2 + 1)2 2(s + 1) 3.11 ((s + 1)2 + 1)2 3.14 1-(1+s)e-• s 2( 1- e-• ) 3.17 (a) 1- e-t (b) 3tet 3.18 (a) e-t -1 + t c) -!(1- (2t + 1)e- 2t) (b) 1- cosbt 3.19 (a) f(t) = + H(t -1) -2t -3t (c) e - e (b) f(t) = (e 2