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E-raamat: Theory of Function Spaces III

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  • Formaat: PDF+DRM
  • Sari: Monographs in Mathematics 100
  • Ilmumisaeg: 10-Sep-2006
  • Kirjastus: Birkhauser Verlag AG
  • Keel: eng
  • ISBN-13: 9783764375829
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Theory of Function Spaces IIISuurem pilt
  • Formaat: PDF+DRM
  • Sari: Monographs in Mathematics 100
  • Ilmumisaeg: 10-Sep-2006
  • Kirjastus: Birkhauser Verlag AG
  • Keel: eng
  • ISBN-13: 9783764375829
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This book may be considered as the continuation of the monographs [ Tri?]and [ Tri?] with the same title. It deals with the theory of function spaces of type s s B and F as it stands at the beginning of this century. These two scales of pq pq spacescovermanywell-knownspacesoffunctionsanddistributionssuchasH¨ older- Zygmundspaces,(fractionalandclassical)Sobolevspaces,BesovspacesandHardy spaces. On the one hand this book is essentially self-contained. On the other hand we concentrate principally on those developments in recent times which are related to the nowadays numerous applications of function spaces to some neighboring areas such as numerics, signal processing and fractal analysis, to mention only a few of them. Chapter 1 in [ Tri?] is a self-contained historically-oriented survey of the function spaces considered and their roots up to the beginning of the 1990s entitled How to measure smoothness. Chapter 1 of the present book has the same heading indicating continuity. As far as the history is concerned we will now be very brief, restricting ourselves to the essentials needed to make this book self-contained and readable. We complement [ Tri?], Chapter 1, by a few points omitted there. But otherwise we jump to the 1990s, describing more recent developments. Some of them will be treated later on in detail.

Arvustused

From the reviews:

"This book can be considered the third volume in an impressive series of books on theory of function spacesbut at the same time it is quite self-containedThe book isextremely well-written, and reader-friendly, and it contains an enormous amount of deep and interesting material. It is strongly recommended to anybody interested in function spaces or in any of the related areas."   Mathematical Reviews

"This book is to be considered as the continuation of the author's two monographs [ Theory of function spaces' [ and] Theory of function spaces. II']; however, it is essentially self-contained and comprehensiveThe author's other two monographs have already yielded deep influence to the development of the theory of function spaces. It is reasonable to expect that the present book will also push this field further. It should be extremely useful to graduate students and experts in the fields of fractal analysis, signal processing, numerics, harmonic analysis, PDE, real analysis, approximation theory and functional analysis."   Zentralblatt MATH

This book is the third part in a famous series of books on the theory of function spaces by the same author. After an intriguing historical survey in the introductory chapter, the book presents thorough discussions of typical building blocks as non-smooth atoms, quarks, wavelet bases, and wavelet frames. The current volume is an impressive and comprehensive account of the state of the art and can to a large extent be read independently . (G. Hörmann, Monatshefte für Mathematik, Vol. 155 (2), October, 2008)

List of Figures x
Preface xi
1 How to Measure Smoothness
1.1 Introduction
1(1)
1.2 Concrete spaces
2(2)
1.3 The Fourier-analytical approach
4(5)
1.4 Local means
9(3)
1.5 Atoms
12(7)
1.5.1 Smooth atoms
12(3)
1.5.2 Non-smooth atoms
15(3)
1.5.3 A technical modification
18(1)
1.6 Quarks
19(7)
1.7 Wavelet bases
26(9)
1.7.1 Mult iresolut ion analysis
26(2)
1.7.2 Haar wavelets
28(2)
1.7.3 Smooth wavelets
30(5)
1.8 Wavelet frames
35(4)
1.9 Envelopes
39(16)
1.9.1 Sharp embeddings
40(5)
1.9.2 The critical case
45(3)
1.9.3 The super-critical case
48(3)
1.9.4 The sub-critical case
51(1)
1.9.5 Some generalisations and further references
52(3)
1.10 Compactness in quasi-Banach spaces
55(3)
1.11 Function spaces on domains
58(20)
1.11.1 General domains: definitions, embeddings
58(2)
1.11.2 General domains: entropy numbers
60(1)
1.11.3 General domains: atoms
61(2)
1.11.4 Lipschitz domains: definitions
63(1)
1.11.5 Lipschitz domains: extension
64(2)
1.11.6 Lipschitz domains: subspaces
66(1)
1.11.7 Lipschitz domains: approximation numbers
67(2)
1.11.8 Lipschitz domains: interpolation
69(3)
1.11.9 Characterisations by differences
72(4)
1.11.10 Lipschitz domains: Sobolev, Holder-Zygmund spaces
76(1)
1.11.11 General domains: sharp embeddings and envelopes
77(1)
1.12 Fractal measures
78(7)
1.12.1 An introduction to the non-smooth
78(2)
1.12.2 Radon measures
80(1)
1.12.3 The ii-property
81(4)
1.13 Fractal operators
85(7)
1.13.1 The classical theory
85(2)
1.13.2 The fractal theory
87(5)
1.14 Fractal characteristics of measures
92(3)
1.15 Isotropic measures
95(4)
1.15.1 Some notation and basic assertions
95(2)
1.15.2 Traces and fractal operators
97(2)
1.16 Weyl measures
99(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
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