Fourier series and wavelets:
Saved in:
| Main Author: | |
|---|---|
| Format: | Book |
| Language: | English |
| Published: |
Amsterdam
Gordon and Breach
1995
|
| Series: | Studies in the development of modern mathematics
3 |
| Subjects: | |
| Online Access: | Inhaltsverzeichnis |
| Physical Description: | XII, 394 S. |
| ISBN: | 2881249930 |
Staff View
MARC
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Record in the Search Index
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CONTENTS
Preface xi
PART I. FOURIER SERIES
Introduction. What are Fourier series about? 1
Chapter 1. Who was Fourier? 3
Chapter 2. The beginning of Fourier series 9
r
1. The analytical theory of heat. Introduction 9
2. Chapters I to III 10
3. Chapters IV to IX 13
4. Back to the introduction 14
Chapter 3. Predecessors and challengers 23
1. The prehistory of harmonic analysis 23
2. Vibrating strings, D. Bernoulli, Euler, and d'Alembert 23
3. Lagrange 25
4. Euler and Fourier formulas, Clairaut 27
5. Poisson and Cauchy 28
6. For further information 29
Chapter 4. Dirichlet and the convergence problem 31
1. Dirichlet 31
2. Comments on the article 31
3. The convergence problem since then 33
4. Dirichlet and Jordan 36
5. Dirichlet's original paper 36
6. A quotation of Jacobi 46
Chapter 5. Riemann and real analysis 49
1. Riemann 49
2. The memoir on trigonometric series. The historical part 50
3. The memoir on trigonometric series. The notion of integral 51
4. The memoir on trigonometric series. Functions representable
by such series 52
5. The memoir on trigonometric series. The final section 54
V
vi CONTENTS
6. Other special trigonometric series. Riemann and Weierstrass 57
7. An overview on the influence of Riemann's memoir just
after 1867 58
8. A partial view on the influence of Riemann's memoir in the
twentieth century 59
9. An excerpt from Riemann's memoir 61
Chapter 6. Cantor and set theory 67
1. Cantor 67
2. Cantor's works on trigonometric series 68
3. Uber die Ausdehnung 69
4. Sets of uniqueness and sets of multiplicity 70
5. Two methods for thin sets in Fourier analysis 72
6. Baire's method 73
7. Randomization 74
8. Another look on Baire's theory 74
9. Recent results and new methods from general set theory 76
10. The first paper in the theory of sets 77
Chapter 7. The turn of the century and Fejer's theorem 87
1. Trigonometric series as a disreputable subject 87
2. The circumstances of Fejer's theorem 89
3. A few applications and continuations of Fejer's theorem 91
Chapter 8. Lebesgue and functional analysis 95
1. Lebesgue 95
2. Lebesgue and Fatou on trigonometric series (1902 1906) 96
3. Trigonometric series and the Lebesgue integral 98
4. Fatou Parseval and Riesz Fischer 99
5. Riesz Fischer and the beginning of Hilbert spaces 100
6. IP, Iq, functions and coefficients 101
7. IP, Hp, conjugate functions 103
8. Functionals 105
9. Approximation 106
Chapter 9. Lacunarity and randomness 109
1. A brief history 109
2. Rademacher, Steinhaus and Gaussian series 111
3. Hadamard series, Riesz products and Sidon sets 113
4. Random trigonometric series 115
5. Application of random methods to Sidon sets 118
CONTENTS vii
6. Lacunary orthogonal series. A(s) sets 120
7. Local and global properties of random trigonometric series 121
8. Local and global properties of lacunary trigonometric series 122
9. Local and global properties of Hadamard trigonometric series 124
Chapter 10. Algebraic structures 127
1. An inheritance from Norbert Wiener 127
2. Compact Abelian groups 128
3. The Wiener Levy theorem 130
4. The converse of Wiener Levy's theorem 132
5. A problem on spectral synthesis with a negative solution 133
6. Another negative result on spectral synthesis 135
7. Homomorphisms of algebras A(G) 136
Chapter 11. Martingales and Hp spaces 139
1. Taylor series, Walsh series, and martingales 139
2. A typical use of Walsh expansions: a best possible Khintchin
inequality 139
3. Walsh series and dyadic martingales 141
4. The Paley theorem on Walsh series 142
5. The W spaces of dyadic martingales 145
6. The classical Hp spaces and Brownian motion 147
Chapter 12. A few classical applications 149
1. Back to Fourier 149
2. The three typical PDEs 149
3. Two extremal problems on curves 151
4. The Poisson formula and the Shannon sampling 154
5. Fast Fourier transform 156
References 159
Index 170
PART II. WAVELETS
Chapter 0. Wavelets: A brief historical account 177
1. Jean Morlet and the beginning of wavelet theory (1982) 177
2. Alex Grossmann and the Marseille team (1984) 180
3. Yves Meyer and the triumph of harmonic analysis (1985) 182
4. Stephane Mallat and the fast wavelet transform (1986) 183
5. Ingrid Daubechies and the FIR filters (1987) 185
viii CONTENTS
Chapter 1. The notion of wavelet representation 189
1. Time frequency localization and Heisenberg's inequality 189
2. Almost orthogonal families, frames and bases in a Hilbert space 193
3. Fourier windows, Gabor wavelets and the Balian Low theorem 196
4. Morlet wavelets 198
5. Wavelet analysis of global regularity 201
6. Wavelet analysis of pointwise regularity 207
Chapter 2. Discrete wavelet transforms 213
1. Sampling theorems for the Morlet wavelet representation 213
2. The vaguelettes lemma and related results for H£ £, spaces 215
3. Proof of the regular sampling theorem 219
4. Proof of the irregular sampling theorem 224
5. Some remarks on dual frames 226
6. Wavelet theory and modern Littlewood Paley theory 228
Chapter 3. The structure of a wavelet basis 231
1. General properties of shift invariant spaces 231
2. The structure of a wavelet basis 240
3. Definition and examples of multi resolution analysis 246
4. Non existence of regular wavelets for the Hardy space H{2) 248
Chapter 4. The theory of scaling filters 253
1. Multi resolution analysis, scaling functions and scaling filters 253
2. Properties of the scaling filters 255
3. Derivatives and primitives of a regular scaling function 264
4. Compactly supported scaling functions 269
Chapter 5. Daubechies' functions and other examples of
scaling functions 277
1. Interpolating scaling functions 277
2. Orthogonal multi resolution analyses 285
3. Spline functions: the case of orthogonal spline wavelets 299
4. Bi orthogonal spline wavelets 303
Chapter 6. Wavelets and functional analysis 307
1. Bi orthogonal wavelets and functional analysis 307
2. Wavelets and Lebesgue spaces 312
3. //' and BMO 320
4. Weighted Lebesgue spaces 326
5. Besov spaces 328
6. Local analysis 335
CONTENTS ix
Chapter 7. Multivariate wavelets 337
1. Multivariate wavelets: a general description 337
2. Existence of multivariate wavelets 341
3. Properties of multivariate wavelets 345
Chapter 8. Algorithms 347
1. The continuous wavelet transform 347
2. Mallat's algorithm 348
3. Wavelets on the interval 353
4. Quadrature formulas 358
5. The BCR algorithm 360
6. The wavelet shrinkage 362
Chapter 9. Further extensions of wavelet theory 363
1. Multiple scaling functions 363
2. Wavelet packets 363
3. Local sine bases 367
4. The matching pursuit algorithm 370
Chapter 10. Some examples of applications of wavelets to analysis 373
1. Wavelets and para products 373
2. The div curl theorem 376
3. Calderon Zygmund operators 378
4. The Riemann function 382
References 385
Index 390 |
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| physical | XII, 394 S. |
| publishDate | 1995 |
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| series | Studies in the development of modern mathematics |
| series2 | Studies in the development of modern mathematics |
| spelling | Kahane, Jean-Pierre 1926-2017 Verfasser (DE-588)131512579 aut Fourier series and wavelets Jean-Pierre Kahane and Pierre-Gilles Lemarié-Rieusset Amsterdam Gordon and Breach 1995 XII, 394 S. txt rdacontent n rdamedia nc rdacarrier Studies in the development of modern mathematics 3 Fourier-Reihe (DE-588)4155109-6 gnd rswk-swf Geschichte (DE-588)4020517-4 gnd rswk-swf Wavelet (DE-588)4215427-3 gnd rswk-swf Fourier-Reihe (DE-588)4155109-6 s DE-604 Wavelet (DE-588)4215427-3 s Geschichte (DE-588)4020517-4 s Lemarié-Rieusset, Pierre-Gilles Sonstige oth Studies in the development of modern mathematics 3 (DE-604)BV003668118 3 HBZ Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=007268039&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Kahane, Jean-Pierre 1926-2017 Fourier series and wavelets Studies in the development of modern mathematics Fourier-Reihe (DE-588)4155109-6 gnd Geschichte (DE-588)4020517-4 gnd Wavelet (DE-588)4215427-3 gnd |
| subject_GND | (DE-588)4155109-6 (DE-588)4020517-4 (DE-588)4215427-3 |
| title | Fourier series and wavelets |
| title_auth | Fourier series and wavelets |
| title_exact_search | Fourier series and wavelets |
| title_full | Fourier series and wavelets Jean-Pierre Kahane and Pierre-Gilles Lemarié-Rieusset |
| title_fullStr | Fourier series and wavelets Jean-Pierre Kahane and Pierre-Gilles Lemarié-Rieusset |
| title_full_unstemmed | Fourier series and wavelets Jean-Pierre Kahane and Pierre-Gilles Lemarié-Rieusset |
| title_short | Fourier series and wavelets |
| title_sort | fourier series and wavelets |
| topic | Fourier-Reihe (DE-588)4155109-6 gnd Geschichte (DE-588)4020517-4 gnd Wavelet (DE-588)4215427-3 gnd |
| topic_facet | Fourier-Reihe Geschichte Wavelet |
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