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Theory of function spaces. III. (English) Zbl 1104.46001
Monographs in Mathematics 100. Basel: Birkhäuser (ISBN 3-7643-7581-7/hbk). xii, 426 p. (2006).
This book is to be considered as the continuation of the author’s two monographs [“Theory of function spaces” (Mathematik und ihre Anwendungen in Physik und Technik 38, Akademische Verlagsgesellschaft Geest & Portig, Leipzig) (1983; Zbl 0546.46028); “Theory of function spaces. II” (Monographs in Mathematics 84, Birkhäuser Verlag, Basel etc.) (1992; Zbl 0763.46025)]; however, it is essentially self-contained and comprehensive. This book deals with the theory of function spaces of type \(B^s_{pq}\) and \(F^s_{pq}\), which is mainly based on the author’s outstanding contributions in this field over the recent years. It is well-known that these two scales of spaces cover many well-known spaces of functions and distributions such as Hölder–Zygmund spaces, Sobolev spaces, Besov spaces, and Hardy spaces. The present book concentrates mainly on those developments in recent times which are related to the nowadays numerous applications of function spaces to numerics, signal processing and fractal analysis.
The book has 9 chapters. Chapter 1 contains a description of the theory of function spaces from the 1990s to now. Yet, the author restricts himself to the essentials needed to make this book self-contained and readable. Chapters 2 and 3 deal with building blocks in (isotropic) spaces of type \(B^s_{pq}\) and \(F^s_{pq}\) in \({\mathbb R}^n\), especially with (nonsmooth) atoms (Chapter 2) and with wavelet bases and wavelet frames (Chapter 3). The author discusses some consequences: pointwise multiplier assertions, positivity properties and local smoothness problems. Chapter 4 is devoted to the theory of function spaces in (bounded) Lipschitz domains in \({\mathbb R}^n\). The author treats wavelet representations of anisotropic function spaces and of weighted function spaces on \({\mathbb R}^n\), respectively, in Chapters 5 and 6. Chapter 7 studies fractal quantities of measures and spectral assertions of fractal elliptic operators. Finally, in Chapters 8 and 9, the author develops a new theory for function spaces on quasimetric spaces and on sets. In short, this book is well-written and its content is rich and well organized.
The 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.

MSC:
46-02 Research exposition (monographs, survey articles) pertaining to functional analysis
46E35 Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems
28A80 Fractals
35P15 Estimates of eigenvalues in context of PDEs
42B35 Function spaces arising in harmonic analysis
42C40 Nontrigonometric harmonic analysis involving wavelets and other special systems
47B06 Riesz operators; eigenvalue distributions; approximation numbers, \(s\)-numbers, Kolmogorov numbers, entropy numbers, etc. of operators
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