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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 | |
algorithm, method, 219
analysis, Gabor, 186
analysis, multiresolution, 26, 381
analysis, multiresolution,
anisotropic, 249
analysis, time-frequency, 186
anisotropy, 237
atom, 12
atom, anisotropic, 245, 375
atom, boundary, 215
atom, 350
atom, (s,p)*, 352
atom, (s,p)*-£, 352
atom, 350, 369
atom, (s,p)k,l, 12
atom, (s,p)%L, 245
atom, 15
atom, 130
atom, (s,p)cr, 63
atom, (s,p)a, 16, 131
ball means, 72, 74, 387
basis, Haar, 29
basis, Schauder, 28
basis, unconditional, 29
Besicovitch, 78
characterisations by differences, 72
characteristics, Besov, 93, 101, 121,
260, 314, 323
characteristics, Bessel, 339
characteristics, Courant, 93
characteristics, multifractal, 93, 101,
314, 323
class, Muckenhoupt, 273
condition, ball, 146
condition, porosity, 146
cone, positive, 84
convergence, local, 153
convergence, unconditional, 14, 34,
153, 270
convergence, absolute, 133, 17
Courant, R., 87
curve, Koch, 359
curve, snowflake, 359
d-set, 91, 96, 313, 315, 357
(d, 4r)-set, 97
d-space, 113, 358, 363
d-space, regular, 358, 373
decomposition, Whitney, 208, 209
Deutsche Mark, Zehn, 125
differences, iterated, 198
differences, symmetric, 319
dimension, Fourier, 303
dimension, Hausdorff, 121
distance, anisotropic, 241
distribution, radial, 288
distribution, tempered, 3, 4, 127
domain in Rn, 58, 194
domain, interior regular, 393
domain, Lipschitz, 64, 195
embedding, compact, non-compact,
197 (proof) embedding, constants, 231
embedding, limiting, 284
embedding, sharp, 58, 77, 40, 43, 258
embedding, weighted, 266, 279
