Wavelets and Operators

Cover......Page 1
Title Page......Page 4
Copyright page......Page 5
Contents......Page 6
Preface to the English Edition......Page 11
Introduction......Page 12
1 Fourier series and integrals, filtering and sampling......Page 18
1 Introduction......Page 18
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
2 Multiresolution approximations of L^2(R^n)......Page 35
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
......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
3 Orthonormal wavelet bases......Page 83
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
4 Non-orthogonal wavelets......Page 147
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
5 Wavelets, the Hardy space Hl and its dual BMO......Page 155
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
6 Wavelets and spaces of functions and distributions......Page 180
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 04 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