Function spaces and wavelets on domains /

Table Of Contents:
Preface v
1 Spaces on Rn and Tn 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 [Rn 13

1.2.1 Wavelets in L2(Rn) 13

1.2.2 Wavelets in Aspq(Rn) 14

1.2.3 Wavelets in Aspq (Rn w) 17

1.3 Periodic spaces on Rn and Tn 19

1.3.1 Definitions and basic properties 19

1.3.2 Wavelets in Apqs,per(Rn) 23

1.3.3 Wavelets in Aspq(Tn) 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 F;pq(Ω) 43

2.3.3 Wavelet para-bases in Lp(Ω), 1<p<infinity 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 Fpq(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 Aspq(Ω) 77

3.2.1 The spaces Fspq(Ω) 77

3.2.2 The spaces Aspq(Ω) I 79

3.2.3 Complemented subspaces 83

3.2.4 Porosity and smoothness zero 85

3.2.5 The spaces Aspq(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,rlocpq(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 Cinfinity 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 (epsilon, 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 Bspq(Rn) 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