The Gibbs Phenomenon in Fourier Analysis, Splines and Wavelet Approximations Mathematics and Its Applications Managing Editor: M.HAZEWINKEL Centre/or Mathematics and Computer Science, Amsterdam, The Netherlands Volume 446 The Gibbs Phenomenon in Fourier Analysis, Splines and Wavelet Approximations by Abdul J. Jerri Department of Mathematics and Computer Science, Clarkson University, Potsdam, New York, U.S.A. Springer·Science+Business Media, B.V A C.I.P. Catalogue record for this book is available from the Library of Congress. ISBN 978-1-4419-4800-7 ISBN 978-1-4757-2847-7 (eBook) DOI 10.1007/978-1-4757-2847-7 Softcover reprint of the hardcover 1st edition 1998 Printed on acid-free paper All Rights Reserved © 1998 Springer Science+Business Media Dordrecht Originally published by Kluwer Academic Publishers in 1998 . No part of the material protected by this copyright notice may be reproduced or utilized in any form or by any means, electronic or mechanical, including photocopying, recording or by any information storage and retrieval system, without written permission from the copyright owner To my caring wife Suad with gratitude Contents Preface xiii ACKNOWLEDGEMENTS xvii AIM OF THE BOOK xix 1 INTRODUCTION 1 1.1 The Gibbs-Wilbraham Phenomenon 1 1.2 Some Basic Elements of Fourier Analysis. 3 1.3 Illustrations and Analysis . 12 A. The Truncated Fourier Series Approximation 12 B. The Truncated Fourier Integral Approximation 16 1.4 Filtering via the Fejer Averaging 26 A. The Fejer Averaging ..... 28 B. The (C, a) Summability ... 31 1.5 The Lanczos-Local-Type Filtering 34 2 ANALYSIS AND FILTERING 37 2.1 The Truncated Fourier Integral . .. 38 2.2 The Fourier Trigonometric Polynomial. .. 40 A. A Note Concerning the General Orthogonal Expansion 43 2.3 The Two Basic Methods of Filtering . 44 A. The Lanczos-Local-Type u-Averaging . 45 B. The Method of Fejer Averaging and Summability 55 2.4 Transform Methods of Filtering . 56 A. The Gegenbauer Transform Method for the Truncated Fourier Series . 57 B. The. Truncated Fourier Integrals . 67 2.5 Examples of Other Filters . 73 The Fourier Series in Two Dimensions 78 2.6 Some Advantages for Edge Detection. 80 vii viii CONTENTS 2.7 A Historical Note ....... 83 2.8 The Higher Dimensional Case. .100 3 THE GENERAL ORTHOGONAL EXPANSIONS 107 3.1 A Brief Overview. 107 3.2 Orthogonal Series Expansions ....... .109 A. The Sturm-Liouville Problem ..... 110 B. The Fourier-Jn-Bessel Series Expansion. 112 C. The Hankel Transform of Radially Symmetric Func- tions in n Dimensions . .. 119 D. The Classical Orthogonal Polynomials Expansion . 122 E. The Legendre Polynomials Series . 122 F. The Tchebychev Polynomials Series . 125 G. The Laguerre Polynomials Series . 127 H. The Hermite Polynomials Series . 130 3.3 The Asymptotic Relation to Fourier Series . 131 Rate of Convergence of the Sturm-Liouville Eigenfunc- tions Expansion. 137 Singular Sturm-Liouville Problem. 140 3.4 The Global Effect on the Convergence in Rn . ....... 148 A. The Laplacian in n-Dimensional-Fourier Series of Ra- dial Functions. 148 B. The 3-Dimensional Case ................. 150 C. the Fourier Integral Representation in n-Dimensions . 155 3.5 Filtering for Orthogonal Expansions . 156 A. The Fejer Averaging ....... 156 B. A Lanczos-Like a-Factor for General Orthogonal Ex- pansions ........................ 157 1. A Lanczos-Like a-Factor for Fourier-Jm-Bessel Series . 159 2. Orthogonal Polynomials Expansions . 171 3. Integral Transforms Representations. 177 4 SPLINES AND OTHER APPROXIMATIONS 183 4.1 The Piecewise-Linear Approximation. 184 4.2 High Order Splines Approximation . 191 4.3 Approximation in Lp-Sense . 199 4.4 The Interpolation of the DFT . 203 5 THE WAVELET REPRESENTATIONS 207 5.1 Wavelets and Fourier Analysis ................ 207 A. The Possible Reason Behind the Gibbs Phenomenon . 207 CONTENTS ix B. Illustration of Some Basic Wavelets, their Fourier Trans­ forms and a Glimpse at the Gibbs Phenomenon .. 216 5.2 Elements of Wavelet Analysis . 222 A. The Continuous Wavelet (Double Integral) Represen­ tation of Functions . 222 B. The Discrete Wavelet (Double) Series Expansion of Functions . 227 5.3 The Discrete Wavelet Series Approximation ........ 230 A. Preliminaries for Having Discrete Orthonormal Wavelets231 5.4 The Continuous Wavelet Representation . 246 A. Detailed Analysis of the Gibbs Phenomenon for Even Wavelets - The Mexican Hat Wavelet ....... 247 B. The Mexican Hat Wavelet and its Gibbs Phenomenon 266 C. Hardy-Functions Wavelets . 274 D. Recent Preliminary Results ............... 285 REFERENCES 287 Appendix A 297 INDEX OF NOTATIONS 319 SUBJECT INDEX 327 AUTHOR INDEX 335 List of Figures 1.1. The square wave function, period 27r. (Many of the Fig- ures and their basic analysis here are from Jerri [11], Inte- gral and Discrete Transforms with Applications and Error Analysis, 1992. Courtesy of Marcel Dekker Inc. .. 13 1.2. The Gibbs phenomenon of the Fourier series partial sum SN(X) of the square wave. N = 10 . .. 13 1.3. A sawtooth function of (1.35) with its clear jump discon­ tinuities. (This and Figs. 1.4, 1.5, 1.7-1.11, and Figs. 2.2-2.4 are from Jerri [11]. Courtesy of Marcel Dekker Inc.) 15 1.4. The initial appearance of the Gibbs phenomenon for the Fourier series approximation of the sawtooth function in (1.35) near the jump discontinuity at x = 2. N = 1,3,5. 15 1.5. The signum function sgn(t), -00 < t < 00. .. 16 1.6. The Gibbs phenomenon of the Fourier integral approxi­ mation sgnB(t) of the signum function sgn(t). B = 40 ... 17 1.7. The sine integral Si(t) and the essence of the Gibbs phe- nomenon near the jump discontinuity at t = O. ...... 18 1.8. Basic functions with jump discontinuities in the interior of their domains. (a) sgn(t), (b) u(t), (c) Pa(t). 20 1.9. A function with a jump discontinuity expressed with the aid of the shifted unit step function. (a) f(t) = 9c(t) + Ju(t - to). (b) 9c(t). (c) u(t - t~). ...... .. 21 1.10. Truncation by the gate function window. The windowing effect: Gb(J) approximation of G(J). 22 1.11. The effect of increasing the width of the window for a continuous input signal. (a) Small window width, (b) Large window width. 24 1.12. The Gibbs phenomenon of sn(x) in (1.57) of approximat- ingthe square wave (see (1.34)) and the Fejer <;l.veraging Sn(x) of (1.63) (or (1.66) for reducing it. n = 40. .. 31 xi xii LIST OF FIGURES 1.5. The signum function sgn(t), -00 < t < 00. 38 1.6 The Gibbs phenomenon of the Fourier integral approxima- tion of sgnB(t) of the signum function sgnB(t). B = 20. 39 1.7. The Sine integral Si(t). .. 39 2.1. The square wave, its Fourier series approximation and the Gibbs phenomenon, N = 10 (the same as Fig. 1.2) .... , 40 2.2. The Gibbs phenomenon of the square wave. N = 10. (This and Figs. 2.3, 2.4 are from Jerri [11]. Courtesy of Marcel Dekker Inc.) ...................... 42 2.3. A possible remedy for the Gibbs phenomenon-approximating sgn(t) by a continuous function sgnul (t). 45 2.4a. The Gibbs phenomenon of the square wave with its (con­ tinuous) 0'1 (averaging) remedy. N = 10. 52 2.4b. The square wave with the 0'1 averaging of the Gibbs phenomenon. N = 22. .. 52 2.4c. The Gibbs phenomenon and its remedies of one (ad and two (0'2) averagings, N = 40. ................ 53 2.4d. The 0'1,0'3 and 0'6 averagings of the Fourier series ap- proximation SN(t) of the square wave in (2.3). ...... 54 2.5. A comparison between the Fejer averaging and the Lanczos- local-0'1 averaging of the Gibbs phenomenon. n = 20. 56 2.6. The Gibbs phenomenon of the truncated Fourier series (2.21) of the sawtooth function, and its absence for the Gegenbauer expansion of (2.33). (Reprinted from [42] with kind permission from Elsevier Science. NL Sara Burgerharstraat 25, 1055 KV Amsterdam, The Nether­ lands.) . .. 62 2.7. The sawtooth function of (2.98). (Most of the figures here are from Hewitt and Hewitt [lO,Fig. 2]. Courtesy of Archives for History of Exact Sciences - Springer-Verlag Inc.) . .. 85 2.8. The sawtooth-like function of (2.99) (from Hewitt and Hewitt [10, Fig. 1.]. Courtesy of Archives for History of Exact Sciences - Springer-Verlag Inc.). .. 85 2.9. The square wave function of (2.100) (From Hewitt and Hewitt [10, Fig. 3]. Courtesy of Archives for History of Exact Sciences - Springer-Verlag Inc.) . .. 86 LIST OF FIGURES xiii 2.10. The different behavior of the undershoots (minima) near x = 0 for SN(X) of (2.99). (From Fig. 10 in Hewitt and Hewitt [10, p. 145]. (Courtesy of Archives for History of Exact Sciences - Springer-Verlag Inc.) . .. 87 2.11a,b. Two cases of Wilbrahm's (correct)11illustrations of the overshoots and undershoots. (Figs. 2.11 and 2.12 are from [10]). Courtesy of Archives for History of Exact Sciences - Springer-Verlag Inc.) . .. 88 2.12a,b. Hewitt and Hewitt's (modern) recomputations ofWilbra­ ham's two illustrations in Fig. 2. 11a,b. (From [10]. Cour- tesy of Archives for History of Exact Sciences - Springer­ Verlag Inc.) . 89 2.13. Graphs of the harmonic analyzer of Michelson and Strat­ ton's [4] for SN(X) of the square wave in (2.100) for N = 1,3,5,7,21 and 79. Note the two cases of N = 79,21 for the Gibbs phenomenon. (From Hewitt and Hewitt [10, Fig. 16]. Courtesy of Archives for History of Exact Sciences - Springer-Verlag Inc.). .............. 95 2.14.