Wavelets and Operators

✍ Scribed by Yves Meyer

• Publisher: Cambridge University Press
• Year: 1993
• Tongue: English
• Leaves: 241
• Series: Cambridge Studies in Advanced Mathematics 48
• Category: Library

✦ Synopsis

Over the last two years, wavelet methods have shown themselves to be of considerable use to harmonic analysts and, in particular, advances have been made concerning their applications. The strength of wavelet methods lies in their ability to describe local phenomena more accurately than a traditional expansion in sines and cosines can. Thus, wavelets are ideal in many fields where an approach to transient behaviour is needed, for example, in considering acoustic or seismic signals, or in image processing. Yves Meyer stands the theory of wavelets firmly upon solid ground by basing his book on the fundamental work of Calder?n, Zygmund and their collaborators. For anyone who would like an introduction to wavelets, this book will prove to be a necessary purchase.

✦ Table of Contents

Cover......Page 1
Title Page......Page 4
Copyright page......Page 5
Contents......Page 6
Preface to the English Edition......Page 11
Introduction......Page 12
2 Fourier series......Page 18
3 Fourier integrals......Page 24
4 Filtering and sampling......Page 26
5 "Wavelets" in the work of Lusin and Calderon......Page 31
1 Introduction......Page 35
2 Multiresolution approximation: definition and examples......Page 39
3 Riesz bases and orthonormal bases......Page 42
4 Regularity of the function \phi......Page 46
5 Bernstein's inequalities......Page 47
6 A remarkable identity satisfied by the operator E_j......Page 49
7 Effectiveness of a multiresolution approximation......Page 58
8 The operators D_j=E_{j+1}-E_j, j€Z......Page 62
9 Besov spaces......Page 67
10 The operators E_j and pseudo-differential operators......Page 71
11 Multiresolution approximations and finite elements......Page 74
12 Example: the Littlewood-paley multiresolution approximation......Page 77
13 Notes and comments......Page 79
1 Introduction......Page 83
2 The construction of wavelets in dimension 1......Page 88
3 Construction of wavelets in dimension 2 by the tensor product method......Page 98
4 The algorithm for constructing multi-dimensional wavelets......Page 100
5 Computing two-dimensional wavelets......Page 104
6 The general existence theorem for wavelet bases......Page 110
7 Cancellation of wavelets......Page 112
8 Wavelets with compact support......Page 113
9 Wavelets with compact support in higher dimensions......Page 124
10 Wavelets and spaces of functions and distributions......Page 126
11 Wavelet series and Fourier series......Page 129
12 Notes and comments......Page 141
1 Introduction......Page 147
2 Frames (or "skew structures")......Page 148
3 Ingrid Daubechies' criterion......Page 149
4 Riesz Bases and Lp convergence......Page 150
1 Introduction......Page 155
2 Equivalent definitions of the space Hl(R.n)......Page 158
3 Atomic decomposition at the coefficient level......Page 161
4 Back to earth......Page 165
5 Atoms and molecules......Page 167
6 The space BMO of John and Nirenberg......Page 168
7 Maurey's theorem......Page 173
8 Notes and complementary remarks......Page 174
1 Introduction......Page 180
2 Criteria for belonging to L^p(R^n) and to L^{p,s}(R^n)......Page 181
3 Hardy spaces H^p(R^n ) with 0<p<1......Page 193
4 Holder spaces......Page 195
5 The Beurling algebra......Page 203
6 The hump algebra......Page 206
7 The space generated by special atoms......Page 209
8 The Bloch space B_{\infty}^{0,\infty}......Page 214
9 Characterization of continuous linear operators T : B_1^{0,1}-+B_1^{0,1}......Page 215
10 Wavelets and Besov spaces......Page 216
11 Holomorphic wavelets and Bochkariev's theorem......Page 219
12 Conclusion......Page 224
Bibliography......Page 225
New references on wavelets and their applications......Page 237
Index......Page 238