Wavelets and other orthogonal systems with applications.

*(English)*Zbl 0866.42022
Boca Raton, FL: CRC Press. x, 248 p. (1994).

The book is devoted to various systems of wavelets in one dimension and to their interrelation with other classical orthogonal systems (orthogonal polynomials, trigonometric series, Rademacher functions, Walsh functions, local sine and cosine bases, Haar system, Shannon system). The author analyses advantages and disadvantages of different systems from the theoretical point of view and taking into account numerous possible applications, e.g., in the signal processing. The book gives a good feeling of the place of wavelets in the realm of orthogonal systems.

The book is divided into 13 chapters. The bibliography contains basic references in this area. Each chapter is provided with exercises and problems. The first 7 chapters are expository and cover the following topics: examples of simplest orthogonal series, tempered distributions and Sobolev spaces, multiresolution analysis, convergence and summability of Fourier series, wavelets based on distributions, orthogonal polynomials and other orthogonal systems. The remaining chapters are more specialized and devoted to pointwise convergence of wavelet expansions, sampling theorems, invariance of orthogonal systems under translation and dilation, analytic representations via orthogonal series, applications in statistics and stochastic processes.

The book may be useful for students, entering to the wavelet theory, to engineers, and to researchers working in this field.

The book is divided into 13 chapters. The bibliography contains basic references in this area. Each chapter is provided with exercises and problems. The first 7 chapters are expository and cover the following topics: examples of simplest orthogonal series, tempered distributions and Sobolev spaces, multiresolution analysis, convergence and summability of Fourier series, wavelets based on distributions, orthogonal polynomials and other orthogonal systems. The remaining chapters are more specialized and devoted to pointwise convergence of wavelet expansions, sampling theorems, invariance of orthogonal systems under translation and dilation, analytic representations via orthogonal series, applications in statistics and stochastic processes.

The book may be useful for students, entering to the wavelet theory, to engineers, and to researchers working in this field.

Reviewer: B.Rubin (Jerusalem)

##### MSC:

42C40 | Nontrigonometric harmonic analysis involving wavelets and other special systems |

42-02 | Research exposition (monographs, survey articles) pertaining to harmonic analysis on Euclidean spaces |