Theory of Function Spaces III

Note: Supplemental materials are not guaranteed with Rental or Used book purchases.

ISBN: 9783764375812 | 3764375817
Cover: Hardcover
Copyright: 9/15/2006

Rent

(Recommended)

$116.43

Term

Due

Price

Semester

Dec 13

$122.56

Quarter

Aug 16

$116.43

Short Term

Jul 6

$110.30

*This item is part of an exclusive publisher rental program and requires an additional convenience fee. This fee will be reflected in the shopping cart.
Buy New

Special Order: 1-2 Weeks

$167.45
eBook

Available Instantly

Online: 180 Days

Downloadable: 180 Days

$138.21

This book deals with the recent theory of function spaces as it stands now. Special attention is paid to some developments in the last 10a??15 years which are closely related to the nowadays numerous applications of the theory of function spaces to some neighbouring areas such as numerics, signal processing and fractal analysis. In particular, typical building blocks as (non-smooth) atoms, quarks, wavelet bases and wavelet frames are discussed in detail and applied afterwards to some outstanding problems of the recent theory of function spaces such as a local smoothness theory, fractal measures, fractal analysis, spaces on Lipschitz domains and on quasi-metric spaces. The book is essentially self-contained, although it might also be considered as the continuation of the two previous books of the author with the same title which appeared as volumes 78 and 84 in this book series. It is directed to mathematicians working in analysis, numerics and fractal geometry, and to (theoretical) physicists interested in related subjects such as signal processing.

List of Figures

Preface

1 How to Measure Smoothness

1.1 Introduction

(1)

1.2 Concrete spaces

(2)

1.3 The Fourier-analytical approach

(5)

1.4 Local means

(3)

1.5 Atoms

(7)

1.5.1 Smooth atoms

(3)

1.5.2 Non-smooth atoms

(3)

1.5.3 A technical modification

(1)

1.6 Quarks

(7)

1.7 Wavelet bases

(9)

1.7.1 Mult iresolut ion analysis

(2)

1.7.2 Haar wavelets

(2)

1.7.3 Smooth wavelets

(5)

1.8 Wavelet frames

(4)

1.9 Envelopes

(16)

1.9.1 Sharp embeddings

(5)

1.9.2 The critical case

(3)

1.9.3 The super-critical case

(3)

1.9.4 The sub-critical case

(1)

1.9.5 Some generalisations and further references

(3)

1.10 Compactness in quasi-Banach spaces

(3)

1.11 Function spaces on domains

(20)

1.11.1 General domains: definitions, embeddings

(2)

1.11.2 General domains: entropy numbers

(1)

1.11.3 General domains: atoms

(2)

1.11.4 Lipschitz domains: definitions

(1)

1.11.5 Lipschitz domains: extension

(2)

1.11.6 Lipschitz domains: subspaces

(1)

1.11.7 Lipschitz domains: approximation numbers

(2)

1.11.8 Lipschitz domains: interpolation

(3)

1.11.9 Characterisations by differences

(4)

1.11.10 Lipschitz domains: Sobolev, Holder-Zygmund spaces

(1)

1.11.11 General domains: sharp embeddings and envelopes

(1)

1.12 Fractal measures

(7)

1.12.1 An introduction to the non-smooth

(2)

1.12.2 Radon measures

(1)

1.12.3 The ii-property

(4)

1.13 Fractal operators

(7)

1.13.1 The classical theory

(2)

1.13.2 The fractal theory

(5)

1.14 Fractal characteristics of measures

(3)

1.15 Isotropic measures

(4)

1.15.1 Some notation and basic assertions

(2)

1.15.2 Traces and fractal operators

(2)

1.16 Weyl measures

(2)

1.17 Spaces on fractals and on quasi-metric spaces

101

(20)

1.17.1 Fractal characteristics, revisited

101

(2)

1.17.2 Traces and trace spaces

103

(6)

1.17.3 Quarkonial representations

109

(3)

1.17.4 Quasi-metric spaces

112

(3)

1.17.5 Spaces on quasi-metric spaces

115

(4)

1.17.6 Function spaces on d-spaces

119

(2)

1.18 Fractal characteristics of distributions

121

(1)

1.19 A black sheep becomes the king

122

(5)

2 Atoms and Pointwise Multipliers

2.1 Notation, definitions and basic assertions

127

(4)

2.1.1 An introductory remark

127

(1)

2.1.2 Basic notation

127

(1)

2.1.3 Spaces on Euclidean n-space

128

(2)

2.1.4 Smooth atoms

130

(1)

2.2 Non-smooth atomic decompositions

131

(5)

2.3 Pointwise multipliers and self-similar spaces

136

(11)

2.3.1 Definitions and preliminaries

136

(1)

2.3.2 Uniform and self-similar spaces

137

(3)

2.3.3 Pointwise multipliers

140

(4)

2.3.4 Comments and complements

144

(3)

3 Wavelets

3.1 Wavelet isomorphisms and wavelet bases

147

(14)

3.1.1 Definitions

147

(2)

3.1.2 Some preparations

149

(4)

3.1.3 The main assertion

153

(4)

3.1.4 Two applications

157

(2)

3.1.5 Further wavelet isomorphisms

159

(2)

3.2 Wavelet frames

161

(25)

3.2.1 Preliminaries and definitions

161

(3)

3.2.2 Subatomic decompositions

164

(6)

3.2.3 Wavelet frames for distributions

170

(4)

3.2.4 Wavelet frames for functions

174

(6)

3.2.5 Local smoothness theory

180

(6)

3.3 Complements

186

(7)

3.3.1 Gausslets

186

(4)

3.3.2 Positivity

190

(3)

4 Spaces on Domains, Wavelets, Sampling Numbers

4.1 Spaces on Lipschitz domains

193

(10)

4.1.1 Introduction

193

(1)

4.1.2 Definitions

194

(1)

4.1.3 Further spaces, some embeddings

195

(3)

4.1.4 Intrinsic characterisations

198

(5)

4.2 Wavelet para-bases

203

(15)

4.2.1 Wavelets in Euclidean n-space, revisited

203

(2)

4.2.2 Scaling properties

205

(2)

4.2.3 Refined localisation

207

(2)

4.2.4 Wavelets in domains: positive smoothness

209

(5)

4.2.5 Wavelets in domains: general smoothness

214

(4)

4.3 Sampling numbers

218

(10)

4.3.1 Definitions

218

(2)

4.3.2 Basic properties

220

(2)

4.3.3 Main assertions

222

(6)

4.4 Complements

228

(7)

4.4.1 Relations to other numbers

228

(3)

4.4.2 Embedding constants

231

(4)

5 Anisotropic Function Spaces

5.1 Definitions and basic properties

235

(14)

5.1.1 Introduction

235

(2)

5.1.2 Definitions

237

(2)

5.1.3 Concrete spaces

239

(4)

5.1.4 New developments

243

(1)

5.1.5 Atoms

244

(3)

5.1.6 Local means

247

(1)

5.1.7 A comment on dual pairings

248

(1)

5.2 Wavelets

249

(7)

5.2.1 Anisotropic multiresolution analysis

249

(3)

5.2.2 Main assertions

252

(4)

5.3 The transference method

256

(7)

5.3.1 The method

256

(2)

5.3.2 Embeddings

258

(1)

5.3.3 Entropy numbers

258

(2)

5.3.4 Besov characteristics

260

(3)

6 Weighted Function Spaces

6.1 Definitions and basic properties

263

(5)

6.1.1 Introduction

263

(1)

6.1.2 Definitions

263

(2)

6.1.3 Basic properties

265

(2)

6.1.4 Special cases

267

(1)

6.2 Wavelets

268

(5)

6.3 A digression: Sequence spaces

273

(6)

6.3.1 Basic spaces

273

(3)

6.3.2 Modifications

276

(3)

6.4 Entropy numbers

279

(7)

6.4.1 The main case

279

(5)

6.4.2 The limiting case

284

(2)

6.5 Complements

286

(11)

6.5.1 The transference method

286

(1)

6.5.2 Radial spaces

287

(10)

7 Fractal Analysis

7.1 Measures

297

(16)

7.1.1 Definitions, basic properties

297

(3)

7.1.2 Potentials and Fourier transforms

300

(3)

7.1.3 Traces: general measures

303

(4)

7.1.4 Traces: isotropic measures

307

(6)

7.2 Characteristics

313

(19)

7.2.1 Characteristics of measures

313

(6)

7.2.2 A digression: Adapted local means

319

(3)

7.2.3 Characteristics of distributions

322

(10)

7.3 Operators

332

(13)

7.3.1 Potentials and the regularity of measures

332

(7)

7.3.2 Elliptic operators: general measures

339

(3)

7.3.3 Elliptic operators: isotropic measures

342

(2)

7.3.4 Weyl measures

344

(1)

8 Function Spaces on Quasi-metric Spaces

8.1 Spaces on d-sets

345

(12)

8.1.1 Introduction

345

(1)

8.1.2 Quarkonial characterisations

346

(4)

8.1.3 Atomic characterisations

350

(7)

8.2 Quasi-metric spaces

357

(6)

8.2.1 d-spaces

357

(2)

8.2.2 Snowflaked transforms

359

(4)

8.3 Spaces on d-spaces

363

(8)

8.3.1 Frames

363

(2)

8.3.2 Spaces of positive smoothness

365

(2)

8.3.3 Spaces of arbitrary smoothness

367

(1)

8.3.4 Spaces of restricted smoothness

368

(3)

8.4 Applications

371

(6)

8.4.1 Entropy numbers

371

(1)

8.4.2 Riesz potentials

372

(2)

8.4.3 Anisotropic spaces

374

(3)

9 Function Spaces on Sets

9.1 Introduction and reproducing formula

377

(4)

9.1.1 Introduction

377

(1)

9.1.2 Reproducing formula

377

(4)

9.2 Spaces on Euclidean n-space

381

(9)

9.2.1 Definitions and basic assertions

381

(5)

9.2.2 Properties

386

(4)

9.3 Spaces on sets

390

(7)

9.3.1 Preliminaries and sequence spaces

390

(4)

9.3.2 Function spaces

394

(3)

References

397

(20)

Notational Agreements

417

(2)

Symbols

419

(4)

Index

423

Please wait while the item is added to your bag...

FREE SHIPPING

Amazon no longer offers textbook rentals. We do!

Theory of Function Spaces III

Summary

Table of Contents

Supplemental Materials