The Gibbs Phenomenon in Fourier Analysis, Splines and Wavelet Approximations

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Springer Science & Business Media, Aug 31, 1998 - Mathematics - 336 pages
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This book represents the first attempt at a unified picture for the pres ence of the Gibbs (or Gibbs-Wilbraham) phenomenon in applications, its analysis and the different methods of filtering it out. The analysis and filtering cover the familiar Gibbs phenomenon in Fourier series and integral representations of functions with jump discontinuities. In ad dition it will include other representations, such as general orthogonal series expansions, general integral transforms, splines approximation, and continuous as well as discrete wavelet approximations. The mate rial in this book is presented in a manner accessible to upperclassmen and graduate students in science and engineering, as well as researchers who may face the Gibbs phenomenon in the varied applications that in volve the Fourier and the other approximations of functions with jump discontinuities. Those with more advanced backgrounds in analysis will find basic material, results, and motivations from which they can begin to develop deeper and more general results. We must emphasize that the aim of this book (the first on the sUbject): to satisfy such a diverse audience, is quite difficult. In particular, our detailed derivations and their illustrations for an introductory book may very well sound repeti tive to the experts in the field who are expecting a research monograph. To answer the concern of the researchers, we can only hope that this book will prove helpful as a basic reference for their research papers.
  

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Contents

IV
1
V
3
VI
12
VIII
16
IX
26
X
28
XI
31
XII
34
XXXIX
137
XL
140
XLI
148
XLII
150
XLIII
155
XLIV
156
XLVI
157
XLVII
159

XIII
37
XIV
38
XV
40
XVI
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XVII
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XVIII
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XIX
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XX
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XXI
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XXII
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XXIII
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XXIV
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XXV
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XXVI
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XXVII
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XXVIII
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XXIX
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XXX
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XXXI
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XXXII
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XXXIII
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XXXV
125
XXXVI
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XXXVII
130
XXXVIII
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XLVIII
171
XLIX
177
L
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LI
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LII
191
LIII
199
LIV
203
LV
207
LVI
216
LVII
222
LVIII
227
LIX
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LX
231
LXI
246
LXII
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LXIII
266
LXIV
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LXV
285
LXVI
287
LXVII
297
LXVIII
319
LXIX
327
LXX
335
Copyright

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Page 288 - RG Cooke, Gibbs phenomenon in Fourier-Bessel series and integrals, Proc. London Math. Soc. (2) 27 (1928), 171.
Page 293 - L. Fejér, Einige Sätze, die sich auf das Vorzeichen einer ganzen rationalen Funktion beziehen,
Page 288 - Error Analysis in Application of Generalization of the Sampling Theorem - chapter 7 in Advanced Topics in Shannon Sampling and Interpolation Theory, RJ Marks II, editor, SpringerVerlag, 1992,

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About the author (1998)

ABDUL J. JERRI, PhD, is Professor of Mathematics at Clarkson University, Potsdam, New York.

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