Preface ......................................................... v
1 Spaces on  n and  n .......................................... 1
1.1 Definitions, atoms, and local means ..................... 1
1.1.1 Definitions ...................................... 1
1.1.2 Atoms ............................................ 4
1.1.3 Local means ...................................... 6
1.2 Spaces on n ........................................... 13
1.2.1 Wavelets in L2(n) ............................. 13
1.2.2 Wavelets in Aspg(n) ............................. 14
1.2.3 Wavelets in Aspg(n,ω) ........................... 17
1.3 Periodic spaces on n and n ...................... 19
1.3.1 Definitions and basic properties ................ 19
1.3.2 Wavelets in As,perpg(n) .......................... 23
1.3.3 Wavelets in Aspg(n) ............................. 26
2 Spaces on arbitrary domains ................................. 28
2.1 Basic definitions ...................................... 28
2.1.1 Function spaces ................................. 28
2.1.2 Wavelet systems and sequence spaces ............. 30
2.2 Homogeneity and refined localisation spaces ............ 33
2.2.1 Homogeneity ..................................... 33
2.2.2 Pointwise multipliers ........................... 35
2.2.3 Refined localisation spaces ..................... 36
2.3 Wavelet para-bases ..................................... 41
2.3.1 Some preparations ............................... 41
2.3.2 Wavelet para-bases in Fs,rlocpg(Ω) ................ 43
2.3.3 Wavelet para-bases in Lp(Ω), 1 < p < ∞ .......... 46
2.4 Wavelet bases .......................................... 48
2.4.1 Orthonormal wavelet bases in L2(Ω) .............. 48
2.4.2 Wavelet bases in Lp(Ω) and Fs,rlocpg(Ω) ........... 53
2.5 Complements ............................................ 55
2.5.1 Haar bases ...................................... 55
2.5.2 Wavelet bases in Lorentz and Zygmund spaces ..... 60
2.5.3 Constrained wavelet expansions for Sobolev
spaces .......................................... 65
3 Spaces on thick domains ..................................... 69
3.1 Thick domains .......................................... 69
3.1.1 Introduction .................................... 69
3.1.2 Classes of domains .............................. 69
3.1.3 Properties and examples ......................... 73
3.2 Wavelet bases in Āspg(Ω) ................................ 77
3.2.1 The spaces  spg(Ω) ............................... 77
3.2.2 The spaces Āspg(Ω) I ............................. 79
3.2.3 Complemented subspaces .......................... 83
3.2.4 Porosity and smoothness zero .................... 85
3.2.5 The spaces Āspg(Ω) II ............................ 89
3.3 Homogeneity and refined localisation, revisited ........ 91
3.3.1 Introduction .................................... 91
3.3.2 Homogeneity: Proof of Theorem 2.11 .............. 92
3.3.3 Wavelet bases in Fs,rlocpg(Ω), revisited .......... 95
3.3.4 Duality ......................................... 97
4 The extension problem ...................................... 101
4.1 Introduction and criterion ............................ 101
4.1.1 Introduction ................................... 101
4.1.2 A criterion .................................... 101
4.2 Main assertions ....................................... 103
4.2.1 Positive smoothness ............................ 103
4.2.2 Negative smoothness ............................ 105
4.2.3 Combined smoothness ............................ 106
4.3 Complements ........................................... 108
4.3.1 Interpolation .................................. 108
4.3.2 Constrained wavelet expansions in Lipschitz
domains ........................................ 112
4.3.3 Intrinsic characterisations .................... 117
4.3.4 Compact embeddings ............................. 123
5 Spaces on smooth domains and manifolds ..................... 130
5.1 Wavelet frames and wavelet-friendly extensions ........ 130
5.1.1 Introduction ................................... 130
5.1.2 Wavelet frames on manifolds .................... 132
5.1.3 Wavelet-friendly extensions .................... 139
5.1.4 Decompositions ................................. 147
5.1.5 Wavelet frames in domains ...................... 151
5.2 Wavelet bases: criterion and lower dimensions ......... 158
5.2.1 Wavelet bases on manifolds ..................... 158
5.2.2 A criterion .................................... 160
5.2.3 Wavelet bases on intervals and planar
domains ........................................ 161
5.3 Wavelet bases: higher dimensions ...................... 163
5.3.1 Introduction ................................... 163
5.3.2 Wavelet bases on spheres and balls ............. 164
5.3.3 Wavelet bases in cellular domains and
manifolds ...................................... 167
5.3.4 Wavelet bases in С∞ domains and cellular
domains ........................................ 172
5.4 Wavelet frames, revisited ............................. 174
5.4.1 Wavelet frames in Lipschitz domains ............ 174
5.4.2 Wavelet frames in (ε, δ)-domains ............... 177
6 Complements ................................................ 178
6.1 Spaces on cellular domains ............................ 178
6.1.1 Riesz bases .................................... 178
6.1.2 Basic properties ............................... 181
6.1.3 A model case: traces and extension ............. 185
6.1.4 A model case: approximation, density,
decomposition .................................. 188
6.1.5 Cubes and polyhedrons: traces and extensions ... 192
6.1.6 Cubes and polyhedrons: Riesz bases ............. 196
6.1.7 Cellular domains: Riesz bases .................. 197
6.2 Existence and non-existence of wavelet frames and
bases ................................................. 199
6.2.1 The role of duality, the spaces Bspg(n) ........ 199
6.2.2 The non-existence of Riesz frames in
exceptional spaces ............................. 202
6.2.3 Reinforced spaces .............................. 204
6.2.4 A proposal ..................................... 208
6.3 Greedy bases .......................................... 210
6.3.1 Definitions and basic assertions ............... 210
6.3.2 Greedy Riesz bases ............................. 212
6.4 Dichotomy: traces versus density ...................... 215
6.4.1 Preliminaries .................................. 215
6.4.2 Traces ......................................... 218
6.4.3 Dichotomy ...................................... 220
6.4.4 Negative smoothness ............................ 226
6.4.5 Curiosities .................................... 226
6.4.6 Pointwise evaluation ........................... 228
6.4.7 A comment on sampling numbers .................. 232
6.5 Polynomial reproducing formulas ....................... 237
6.5.1 Global reproducing formulas .................... 237
6.5.2 Local reproducing formulas ..................... 239
6.5.3 A further comment on sampling numbers .......... 240
Bibliography .......................................... 243
Symbols ....................................................... 251
Index ......................................................... 255
|
