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**Wavelets, their friends, and what they can do for you.**
*(English)*
Zbl 1153.42016

EMS Series of Lectures in Mathematics. Zürich: European Mathematical Society (EMS) (ISBN 978-3-03719-018-0/pbk). x, 109 p. (2008).

These Lecture Notes introduce the reader in an informal way to the central topics surrounding wavelets and their applications and they should be very useful as a complement to introductory texts on the subject. The style is focused on ideas, whereas details and proofs are often referred to the existing literature.

The book with its approximately 100 pages consists of six chapters. At first, background material about bases and frames in Banach spaces is shortly presented. Chapter 2 is devoted to Time-frequency analysis. Classical Fourier analysis is shortly reviewed and the fast Fourier transform is given in detail. The windowed Fourier transform and the Wavelet transform are introduced and the Heisenberg uncertainty principle and the theorem of Balian-Low are presented. The section contains also a nice discussion of local cosine bases.

Chapter 3 begins with the concept of multiresolution analysis. Basic results are described, and explained in the setting of the Haar wavelet. The text proceeds to a discussion of the fast wavelet transform, filters and filter banks, Daubechies style wavelets, special properties of wavelets like compact support, smoothness and the construction of further special wavelets.

Chapter 4, titled “Friends, relatives and mutation of wavelets”, is devoted to biorthogonal wavelets, higher order splines wavelets, multiwavelets, Alpert multiwavelets, wavelet packets and wavelets in two dimensions. The reader will find short and informative explanations, and many interesting references are given for the beginning reader. The section then dwells on so-called second generation wavelets: wavelets on the interval, folded wavelets, boundary wavelets, wavelets on bounded domains and several notions of wavelets in signal processing. It finishes with a discussion of prolate spheroidal wave functions.

Chapter 5 is devoted to applications, and as the authors write, without attempting to be systematic or comprehensive, and dictated by the authors’ experience in this area. Basic principles in compression and denoising are explained. It is shown how to calculate derivatives with wavelets. Applications to functional analysis are given: characterizations of function spaces via wavelet coefficients and a discussion of local regularity. The chapter finishes with applications to differential equations: Galerkin methods using wavelets and operator calculus approaches for PDEs.

In Chapter 6 the authors shortly comment on the existing literature and software.

The book is written in a vivid and fresh style and provides for the beginning reader a very good survey about many different developments in the area of wavelet analysis and its applications.

The book with its approximately 100 pages consists of six chapters. At first, background material about bases and frames in Banach spaces is shortly presented. Chapter 2 is devoted to Time-frequency analysis. Classical Fourier analysis is shortly reviewed and the fast Fourier transform is given in detail. The windowed Fourier transform and the Wavelet transform are introduced and the Heisenberg uncertainty principle and the theorem of Balian-Low are presented. The section contains also a nice discussion of local cosine bases.

Chapter 3 begins with the concept of multiresolution analysis. Basic results are described, and explained in the setting of the Haar wavelet. The text proceeds to a discussion of the fast wavelet transform, filters and filter banks, Daubechies style wavelets, special properties of wavelets like compact support, smoothness and the construction of further special wavelets.

Chapter 4, titled “Friends, relatives and mutation of wavelets”, is devoted to biorthogonal wavelets, higher order splines wavelets, multiwavelets, Alpert multiwavelets, wavelet packets and wavelets in two dimensions. The reader will find short and informative explanations, and many interesting references are given for the beginning reader. The section then dwells on so-called second generation wavelets: wavelets on the interval, folded wavelets, boundary wavelets, wavelets on bounded domains and several notions of wavelets in signal processing. It finishes with a discussion of prolate spheroidal wave functions.

Chapter 5 is devoted to applications, and as the authors write, without attempting to be systematic or comprehensive, and dictated by the authors’ experience in this area. Basic principles in compression and denoising are explained. It is shown how to calculate derivatives with wavelets. Applications to functional analysis are given: characterizations of function spaces via wavelet coefficients and a discussion of local regularity. The chapter finishes with applications to differential equations: Galerkin methods using wavelets and operator calculus approaches for PDEs.

In Chapter 6 the authors shortly comment on the existing literature and software.

The book is written in a vivid and fresh style and provides for the beginning reader a very good survey about many different developments in the area of wavelet analysis and its applications.

Reviewer: Hermann Render (Dublin)

### 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 |

65T60 | Numerical methods for wavelets |