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Wavelet analysis. The scalable structure of information. (English) Zbl 0922.42020

New York, NY: Springer. xvi, 435 p. (1998).
The book introduces the ideas and methods lying behind the theory of compactly supported wavelets and illustrates their applications in mathematics and engineering. It splits into the following four parts: Scalable structure of information, Wavelet theory, Wavelet approximation and algorithms, Wavelets applications.
In the first part the multiscale nature of information in the mathematical and natural world is illustrated.
The second part presents the theory of scaling and wavelet functions and the corresponding multiresolution-analysis.
In part 3 it is shown, how finite scale wavelet series can be used to represent a function or a signal.
The fundamental Mallat algorithms are developed. The notion of Coiflets is introduced, and a Newton method is applied for their numerical construction.
A theory of wavelet based differentiation is presented using the concept of connection coefficients.
Further, a wavelet multiscale representation for geometric regions and geometric Euclidean spaces is introduced. The above ideas are applied in order to find multiscale numerical algorithms for scaling elliptic boundary value problems based on the Galerkin-method.
In part 4, the authors illustrate a number of applications of wavelets to problems of data compression and telecommunications.
The book is of high interest for readers being especially interested in the basic ideas of wavelet application in different areas of mathematics and signal processing.

MSC:

42C40 Nontrigonometric harmonic analysis involving wavelets and other special systems
42-01 Introductory exposition (textbooks, tutorial papers, etc.) pertaining to harmonic analysis on Euclidean spaces
94A12 Signal theory (characterization, reconstruction, filtering, etc.)
94-01 Introductory exposition (textbooks, tutorial papers, etc.) pertaining to information and communication theory
65-01 Introductory exposition (textbooks, tutorial papers, etc.) pertaining to numerical analysis
65T99 Numerical methods in Fourier analysis
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