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Fractal functions, fractal surfaces, and wavelets. (English) Zbl 0817.28004

Orlando, FL: Academic Press, Inc. xi, 383 p. (1994).
The first impression about this book is that the author has the intention to introduce also the very inexperienced reader to the main subjects: fractal functions, fractal surfaces, and their interactions with wavelet theory. Thus, it is organized in two parts. Foundations on the one hand and fractal functions and surfaces on the other hand. In the first part the author does it without extended proofs, but he passes indeed through all the theory necessary to understand the advanced topics. It includes rather general mathematical preliminaries like analysis, topology, probability theory, and algebra, but also a considerable systematic account of different constructions of fractal sets (classical examples, IFS, recurrence, and graph directed construction), dimension theory, without incorporating packing dimension, up to dynamical systems. The first two chapters of part two, written for specialists, are devoted to the fractal function constructions and related dimension theory. The key notion is the Read-Bajraktarević operator. The heart of the book is Chapter 7 dealing with the interaction of wavelet theory and fractal functions. Although the detailed examples are very helpful at this place, a clear definition of Daubechies wavelets is missing. Especially, the inexperienced reader will have difficulties to recognize the main advantage of fractal function wavelets, at first reading, since the introductory remarks of this chapter are not related to the results stated later. Chapters on fractal surfaces and the generalization of fractal wavelets to \(\mathbb{R}^ n\) complete the material.

MSC:

28A80 Fractals
42C40 Nontrigonometric harmonic analysis involving wavelets and other special systems
28-01 Introductory exposition (textbooks, tutorial papers, etc.) pertaining to measure and integration
42-01 Introductory exposition (textbooks, tutorial papers, etc.) pertaining to harmonic analysis on Euclidean spaces
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