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| List of Figures | x | |
| Preface | xi | |
| 1 How to Measure Smoothness | ||
| 1.1 | Introduction | 1 |
| 1.2 | Concrete spaces | 2 |
| 1.3 | The Fourier-analytical approach | 4 |
| 1.4 | Local means | 9 |
| 1.5 | Atoms | 12 |
| 1.5.1 | Smooth atoms | 12 |
| 1.5.2 | Non-smooth atoms | 15 |
| 1.5.3 | A technical modification | 18 |
| 1.6 | Quarks | 19 |
| 1.7 | Wavelet bases | 26 |
| 1.7.1 | Multiresolution analysis | 26 |
| 1.7.2 | Haar wavelets | 28 |
| 1.7.3 | Smooth wavelets | 30 |
| 1.8 | Wavelet frames | 35 |
| 1.9 | Envelopes | 39 |
| 1.9.1 | Sharp embeddings | 40 |
| 1.9.2 | The critical case | 45 |
| 1.9.3 | The super-critical case | 48 |
| 1.9.4 | The sub-critical case | 51 |
| 1.9.5 | Some generalisations and further references | 52 |
| 1.10 | Compactness in quasi-Banach spaces | 55 |
| 1.11 | Function spaces on domains | 58 |
| 1.11.1 | General domains: definitions, embeddings | 58 |
| 1.11.2 | General domains: entropy numbers | 60 |
| 1.11.3 | General domains: atoms | 61 |
| 1.11.4 | Lipschitz domains: definitions | 63 |
| 1.11.5 | Lipschitz domains: extension | 64 |
| 1.11.6 | Lipschitz domains: subspaces | 66 |
| 1.11.7 | Lipschitz domains: approximation numbers | 67 |
| 1.11.8 | Lipschitz domains: interpolation | 69 |
| 1.11.9 | Characterisations by differences | 72 |
| 1.11.10 | Lipschitz domains: Sobolev, Holder-Zygmund spaces | 76 |
| 1.11.11 | General domains: sharp embeddings and envelopes | 77 |
| 1.12 | Fractal measures | 78 |
| 1.12.1 | An introduction to the non-smooth | 78 |
| 1.12.2 | Radon measures | 80 |
| 1.12.3 | The ¡i-property | 81 |
| 1.13 | Fractal operators | 85 |
| 1.13.1 | The classical theory | 85 |
| 1.13.2 | The fractal theory | 87 |
| 1.14 | Fractal characteristics of measures | 92 |
| 1.15 | Isotropic measures | 95 |
| 1.15.1 | Some notation and basic assertions | 95 |
| 1.15.2 | Traces and fractal operators | 97 |
| 1.16 | Weyl measures | 99 |
| 1.17 | Spaces on fractals and on quasi-metric spaces | 101 |
| 1.17.1 | Fractal characteristics, revisited | 101 |
| 1.17.2 | Traces and trace spaces | 103 |
| 1.17.3 | Quarkonial representations | 109 |
| 1.17.4 | Quasi-metric spaces | 112 |
| 1.17.5 | Spaces on quasi-metric spaces | 115 |
| 1.17.6 | Function spaces on d-spaces | 119 |
| 1.18 | Fractal characteristics of distributions | 121 |
| 1.19 | A black sheep becomes the king | 122 |
| 2 Atoms and Pointwise Multipliers | ||
| 2.1 | Notation, definitions and basic assertions | 127 |
| 2.1.1 | An introductory remark | 127 |
| 2.1.2 | Basic notation | 127 |
| 2.1.3 | Spaces on Euclidean n-space | 128 |
| 2.1.4 | Smooth atoms | 130 |
| 2.2 | Non-smooth atomic decompositions | 131 |
| 2.3 | Pointwise multipliers and self-similar spaces | 136 |
| 2.3.1 | Definitions and preliminaries | 136 |
| 2.3.2 | Uniform and self-similar spaces | 137 |
| 2.3.3 | Pointwise multipliers | 140 |
| 2.3.4 | Comments and complements | 144 |
| 3 Wavelets | ||
| 3.1 | Wavelet isomorphisms and wavelet bases | 147 |
| 3.1.1 | Definitions | 147 |
| 3.1.2 | Some preparations | 149 |
| 3.1.3 | The main assertion | 153 |
| 3.1.4 | Two applications | 157 |
| 3.1.5 | Further wavelet isomorphisms | 159 |
| 3.2 | Wavelet frames | 161 |
| 3.2.1 | Preliminaries and definitions | 161 |
| 3.2.2 | Subatomic decompositions | 164 |
| 3.2.3 | Wavelet frames for distributions | 170 |
| 3.2.4 | Wavelet frames for functions | 174 |
| 3.2.5 | Local smoothness theory | 180 |
| 3.3 | Complements | 186 |
| 3.3.1 | Gausslets | 186 |
| 3.3.2 | Positivity | 190 |
| 4 Spaces on Domains, Wavelets, Sampling Numbers | ||
| 4.1 | Spaces on Lipschitz domains | 193 |
| 4.1.1 | Introduction | 193 |
| 4.1.2 | Definitions | 194 |
| 4.1.3 | Further spaces, some embeddings | 195 |
| 4.1.4 | Intrinsic characterisations | 198 |
| 4.2 | Wavelet para-bases | 203 |
| 4.2.1 | Wavelets in Euclidean n-space, revisited | 203 |
| 4.2.2 | Scaling properties | 205 |
| 4.2.3 | Refined localisation | 207 |
| 4.2.4 | Wavelets in domains: positive smoothness | 209 |
| 4.2.5 | Wavelets in domains: general smoothness | 214 |
| 4.3 | Sampling numbers | 218 |
| 4.3.1 | Definitions | 218 |
| 4.3.2 | Basic properties | 220 |
| 4.3.3 | Main assertions | 222 |
| 4.4 | Complements | 228 |
| 4.4.1 | Relations to other numbers | 228 |
| 4.4.2 | Embedding constants | 231 |
| 5 Anisotropic Function Spaces | ||
| 5.1 | Definitions and basic properties | 235 |
| 5.1.1 | Introduction | 235 |
| 5.1.2 | Definitions | 237 |
| 5.1.3 | Concrete spaces | 239 |
| 5.1.4 | New developments | 243 |
| 5.1.5 | Atoms | 244 |
| 5.1.6 | Local means | 247 |
| 5.1.7 | A comment on dual pairings | 248 |
| 5.2 | Wavelets | 249 |
| 5.2.1 | Anisotropic multiresolution analysis | 249 |
| 5.2.2 | Main assertions | 252 |
| 5.3 | The transference method | 256 |
| 5.3.1 | The method | 256 |
| 5.3.2 | Embeddings | 258 |
| 5.3.3 | Entropy numbers | 258 |
| 5.3.4 | Besov characteristics | 260 |
| 6 Weighted Function Spaces | ||
| 6.1 | Definitions and basic properties | 263 |
| 6.1.1 | Introduction | 263 |
| 6.1.2 | Definitions | 263 |
| 6.1.3 | Basic properties | 265 |
| 6.1.4 | Special cases | 267 |
| 6.2 | Wavelets | 268 |
| 6.3 | A digression: Sequence spaces | 273 |
| 6.3.1 | Basic spaces | 273 |
| 6.3.2 | Modifications | 276 |
| 6.4 | Entropy numbers | 279 |
| 6.4.1 | The main case | 279 |
| 6.4.2 | The limiting case | 284 |
| 6.5 | Complements | 286 |
| 6.5.1 | The transference method | 286 |
| 6.5.2 | Radial spaces | 287 |
| 7 Fractal Analysis | ||
| 7.1 | Measures | 297 |
| 7.1.1 | Definitions, basic properties | 297 |
| 7.1.2 | Potentials and Fourier transforms | 300 |
| 7.1.3 | Traces: general measures | 303 |
| 7.1.4 | Traces: isotropic measures | 307 |
| 7.2 | Characteristics | 313 |
| 7.2.1 | Characteristics of measures | 313 |
| 7.2.2 | A digression: Adapted local means | 319 |
| 7.2.3 | Characteristics of distributions | 322 |
| 7.3 | Operators | 332 |
| 7.3.1 | Potentials and the regularity of measures | 332 |
| 7.3.2 | Elliptic operators: general measures | 339 |
| 7.3.3 | Elliptic operators: isotropic measures | 342 |
| 7.3.4 | Weyl measures | 344 |
| 8 Function Spaces on Quasi-metric Spaces | ||
| 8.1 | Spaces on d-sets | 345 |
| 8.1.1 | Introduction | 345 |
| 8.1.2 | Quarkonial characterisations | 346 |
| 8.1.3 | Atomic characterisations | 350 |
| 8.2 | Quasi-metric spaces | 357 |
| 8.2.1 | d-spaces | 357 |
| 8.2.2 | Snowflaked transforms | 359 |
| 8.3 | Spaces on d-spaces | 363 |
| 8.3.1 | Frames | 363 |
| 8.3.2 | Spaces of positive smoothness | 365 |
| 8.3.3 | Spaces of arbitrary smoothness | 367 |
| 8.3.4 | Spaces of restricted smoothness | 368 |
| 8.4 | Applications | 371 |
| 8.4.1 | Entropy numbers | 371 |
| 8.4.2 | Riesz potentials | 372 |
| 8.4.3 | Anisotropic spaces | 374 |
| 9 Function Spaces on Sets | ||
| 9.1 | Introduction and reproducing formula | 377 |
| 9.1.1 | Introduction | 377 |
| 9.1.2 | Reproducing formula | 377 |
| 9.2 | Spaces on Euclidean n-space | 381 |
| 9.2.1 | Definitions and basic assertions | 381 |
| 9.2.2 | Properties | 386 |
| 9.3 | Spaces on sets | 390 |
| 9.3.1 | Preliminaries and sequence spaces | 390 |
| 9.3.2 | Function spaces | 394 |
| References | 397 | |
| Notational Agreements | 417 | |
| Symbols | 419 | |
| Index | 423 | |
