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Function spaces and wavelets on domains. (English) Zbl 1158.46002
EMS Tracts in Mathematics 7. Zürich: European Mathematical Society (EMS) (ISBN 978-3-03719-019-7/hbk). ix, 256 p. (2008).
This book may be considered as a continuation of the author’s three monographs [Theory of function spaces, Leipzig: Akademische Verlagsgesellschaft (1983; Zbl 0546.46028); Theory of function spaces. II, Basel etc.: Birkhäuser Verlag (1992; Zbl 0763.46025); Theory of function spaces. III, Basel etc.: Birkhäuser Verlag (2006; Zbl 1104.46001)].
It is well known that wavelets have emerged as an important tool in analyzing functions containing discontinuities and sharp spikes. They were developed independently in the fields of mathematics, quantum physics, electrical engineering, and seismic geology. Interactions between these fields during the last ten years have led to many new wavelet applications such as image compression, turbulence, human vision, radar, earthquake prediction, and pure mathematics applications such as solving partial differential equations.
In this book, the author develops a theory of wavelet bases and wavelet frames for function spaces on various types of domains in $${\mathbb R}^n$$. The author also studies extension problems. In Chapter 1, the author deals with the usual spaces on $${\mathbb R}^n$$, periodic spaces on $${\mathbb R}^n$$ and on the $$n$$-torus $${\mathbb T}^n$$, and their wavelet expansions under natural restrictions for the parameters involved. Spaces on arbitrary domains are the subject of Chapter 2. The heart of the exposition are Chapters 3 and 4, where the author develops a theory of function spaces on so-called thick domains, including wavelet expansions and extensions to corresponding spaces on $${\mathbb R}^n$$. This is complemented in Chapter 5 by spaces on smooth manifolds and smooth domains. Finally, the author adds in Chapter 6 a discussion about desirable properties of wavelet expansions in function spaces, introducing the notation of Riesz wavelet bases and frames. This chapter also deals with some related topics, in particular, with spaces on cellular domains.
Although the presentation of this book relies on the recent theory of function spaces, basic notations and classical results are recalled in order to make the text self-contained. This book is addressed to two types of readers: researchers in the theory of function spaces who are interested in wavelets as new effective building blocks for functions, and scientists who wish to use wavelet bases in classical function spaces for various applications. Adapted to the second type of readers, the preface contains a guide to where one finds basic definitions and key assertions.
In short, as the author’s other monographs, this book is well written and its content is rich and well organized. It should be an extremely useful reference of this field.

##### 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 42C40 Nontrigonometric harmonic analysis involving wavelets and other special systems 42B35 Function spaces arising in harmonic analysis 28A80 Fractals 46N99 Miscellaneous applications of functional analysis
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