The Gibbs Phenomenon in Fourier Analysis, Splines and Wavelet Approximations

340 Pages·1998·4.7 MB·English

by  Abdul J. Jerri (auth.)

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