1996 • 345 Pages • 22.52 MB • English

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To TANYA and LIZ- with love and respect

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Applied and Numerical Harmonic Analysis Distributions in the Physical and Engineering Sciences

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Applied and Numerical Harmonic Analysis Series Editor JOHN J. BENEDETTO University 0/ Maryland Editorial Board Akram Aldroubi Douglas Cochran NIH, Biomedical Engineering/Instrumentation Arizona State University Ingrid Daubechies Hans G. Feichtinger Princeton University University 0/ Vienna Christopher Heil Murat Kunt Georgia Institute o/Technology Swiss Federal Institute o/Technology, Lausanne James McClellan Wim Sweldens Georgia Institute o/Technology Lucent Technologies, Bell Laboratories Michael Unser Martin Vetterli NIH, BiomedicalEngineering/instrumentation Swiss Federal Institute o/Technology, Lausanne Victor Wickerhauser Washington University

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Alexander l. SAICHEV University of Nizhniy Novgorod and Wojbor A. WOYCZYNSKI Case Western Reserve University DISTRIBUTIONS IN THE PHYSICAL AND ENGINEERING SCIENCES Volume 1 Distributional and Fractal Calculus, Integral Transforms and Wavelets BIRKHAUSER Boston Basel Berlin

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Alexander I. Saichev W ojbor A. W oyczynski Radio Physics Department Department of Statistics and Center University of Nizhniy Novgorod for Stochastic and Chaotic Processes Nizhniy Novgorod, 603022 in Science and Technology Russia Case Western Reserve University Cleveland, Ohio 44106 U.S.A. Library of Congress Cataloging In-Publication Data Woyczynski, W. A. (Wojbor Andrzej), 1943- Distributions in the physical and engineering sciences / Wojbor A. Woyczynski, Alexander I. Saichev. p. cm. -- (Applied and numerical harmonic analysis) Includes bibliographical references and index. Contents: V. 1. Distributional and fractal calculus, integral transforms, and wavelets. ISBN-13: 978-1-4612-8679-0 e-ISBN-13: 978-1-4612-4158-4 DOl: 10.1007978-1-4612-4158-4 1. Theory of distributions (Functional analysis) I. Saichev, A. I. II. Title. III. Series. QA324.w69 1996 515'.782' 0245--dc20 96-39028 CIP Printed on acid-free paper m® © 1997 Birkhauser Boston Birkhauser H02' Copyright is not claimed for works of U.S. Government employees. All rights reserved. No part of this publication may be reproduced, stored in a retrieval system, or transmitted, in any form or by any means, electronic, mechanical, photocopying, recording, or otherwise, without prior permission of the copyright owner. Permission to photocopy for internal or personal use of specific clients is granted by Birkhauser Boston for libraries and other users registered with the Copyright Clearance Center (CCC), provided that the base fee of$6.00 per copy, plus $0.20 per page is paid directly to CCC, 222 Rosewood Drive, Danvers, MA 01923, U.S.A. Special requests should be addressed directly to Birkhauser Boston, 675 Massachusetts A venue, Cambridge, MA 02139, U.S.A. ISBN -13: 978-1-4612-8679-0 Camera-ready text prepared in LA1EX by T & T TechWorks Inc., Coral Springs, FL. 987 6 543 2 1

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Contents Introduction xi Notation xvii Part I DISTRIBUTIONS AND THEIR BASIC APPLICATIONS 1 1 Basic Definitions and Operations 3 1.1 The "delta function" as viewed by a physicist and an engineer . 3 1.2 A rigorous definition of distributions . . . . . . . . 5 1.3 Singular distributions as limits of regular functions . 10 1.4 Derivatives; linear operations ............ 14 1.5 Multiplication by a smooth function; Leibniz formula 17 1.6 Integrals of distributions; the Heaviside function . 20 1.7 Distributions of composite arguments . . 24 1.8 Convolution . . . . . . . . . . . . . . . 27 1.9 The Dirac delta on Rn , lines and surfaces 28 1.10 Linear topological space of distributions . 31 1.11 Exercises . . . . . . . . . . . . . . . . . 34 2 Basic Applications: Rigorous and Pragmatic 37 2.1 Two generic physical examples ........... 37 2.2 Systems governed by ordinary differential equations . 39 2.3 One-dimensional waves . . . . . . . . . . . . 43 2.4 Continuity equation . . . . . . . . . . . . . . 44 2.5 Green's function of the continuity equation and Lagrangian coordinates ............ 49 2.6 Method of characteristics ........... 51 2.7 Density and concentration of the passive tracer 54 2.8 Incompressible medium . . . . . . . . . . . . 55

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viii Contents 2.9 Pragmatic applications: beyond the rigorous theory of distributions . 57 2.10 Exercises . . . . . . . . . . . . . . . . . . . . . . 70 Part IT INTEGRAL TRANSFORMS AND DIVERGENT SERIES 73 3 Fourier Transform 75 3.1 Definition and elementary properties . . . . . . 75 3.2 Smoothness, inverse transform and convolution 78 3.3 Generalized Fourier transform 81 3.4 Transport equation 84 3.5 Exercises .......... . 90 4 Asymptotics of Fourier Transforms 93 4.1 Asymptotic notation, or how to get a camel to pass through a needle's eye .. . . . . . . 93 4.2 Riemann-Lebesgue Lemma . . . . . . . . . . . . . . . . . 98 4.3 Functions with jumps ................... . 101 4.4 Gamma function and Fourier transforms of power functions . 112 4.5 Generalized Fourier transforms of power functions 123 4.6 Discontinuities of the second kind 130 4.7 Exercises ............. . 134 5 Stationary Phase and Related Method 137 5.1 Finding asymptotics: a general scheme 137 5.2 Stationary phase method ....... . 140 5.3 Fresnel approximation . . . . . . . . . 141 5.4 Accuracy of the stationary phase method. 142 5.5 Method of steepest descent. 145 5.6 Exercises .............. . 146 6 Singular Integrals and Fractal Calculus 149 6.1 Principal value distribution. . . . 149 6.2 Principal value of Cauchy integral 152 6.3 A study of monochromatic wave . 153 6.4 The Cauchy formula . 157 6.5 The Hilbert transform . . . . . . 160 6.6 Analytic signals . . . . . . . . . 162 6.7 Fourier transform of Heaviside function 163 6.8 Fractal integration .. 166 6.9 Fractal differentiation 170 6.10 Fractal relaxation . 175 6.11 Exercises . . . . . . . 180

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Contents ix 7 Uncertainty Principle and Wavelet Transforms 183 7.1 Functional Hilbert spaces ................ 183 7.2 Time-frequency localization and the uncertainty principle 190 7.3 Windowed Fourier transform. . . . . . . . 193 7.4 Continuous wavelet transforms. . . . . . . 210 7.5 Haar wavelets and multiresolution analysis 225 7.6 Continuous Daubechies' wavelets 231 7.7 Wavelets and distributions 237 7.8 Exercises............. 243 8 Summation of Divergent Series and Integrals 245 8.1 Zeno's "paradox" and convergence of infinite series 245 8.2 Summation of divergent series . . . . . . . . . . . 253 8.3 Tiring Achilles and the principle of infinitesimal relaxation 255 8.4 Achilles chasing the tortoise in presence of head winds 258 8.5 Separation of scales condition . 260 8.6 Series of complex exponentials 264 8.7 Periodic Dirac deltas. . . . . . 268 8.8 Poisson summation formula . . 271 8.9 Summation of divergent geometric series 273 8.10 Shannon's sampling theorem. 276 8.11 Divergent integrals . 281 8.12 Exercises. . . . . . . . . . . 283 A Answers and Solutions 287 A.1 Chapter 1. Definitions and operations 287 A.2 Chapter 2. Basic applications .... 288 A.3 Chapter 3. Fourier transform. . . . . 292 A.4 Chapter 4. Asymptotics of Fourier transforms 294 A.5 Chapter 5. Stationary phase and related methods 296 A.6 Chapter 6. Singular integrals and fractal calculus 302 A.7 Chapter 7. Uncertainty principle and wavelet transform 308 A.8 Chapter 8. Summation of divergent series and integrals 312 B Bibliographical Notes 325 Index 331

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Introduction Goals and audience The usual calculus/differential equations sequence taken by the physical sciences and engineering majors is too crowded to include an in-depth study of many widely applicable mathematical tools which should be a part of the intellectual arsenal of any well educated scientist and engineer. So it is common for the calculus sequence to be followed by elective undergraduate courses in linear algebra, probability and statistics, and by a graduate course that is often labeled Advanced Mathematics for Engineers and Scientists. Traditionally, it contains such core topics as equations of mathematical physics, special functions, and integral transforms. This book is designed as a text for a modern version of such a graduate course and as a reference for theoretical researchers in the physical sciences and engineering. Nevertheless, inasmuch as it contains basic definitions and detailed explanations of a number of traditional and modern mathematical notions, it can be comfortably and profitably taken by advanced undergraduate students. It is written from the unifying viewpoint of distribution theory and enriched by such modern topics as wavelets, nonlinear phenomena and white noise theory, which became very important in the practice of physical scientists. The aim of this text is to give the readers a major modern analytic tool in their research. Students will be able to independently attack problems where distribution theory is of importance. Prerequisites include a typical science or engineering 3-4 semester calculus sequence (including elementary differential equations, Fourier series, complex variables and linear algebra-we review the basic definitions and facts as needed). No probability background is necessary as all the concepts are explained from scratch. In solving some problems, familiarity with basic computer programming methods is necessary although using a symbolic manipulation language such as Mathematica, MATLAB or Maple would suffice. These skills should be acquired during freshman and sophomore years.

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