A first course on wavelets.

*(English)*Zbl 0885.42018
Studies in Advanced Mathematics. Boca Raton, FL: CRC Press. 489 p. (1996).

This book aims to introduce the discrete wavelet theory as a powerful tool in mathematical analysis. It can be seen as a good complement of the well-known books by Meyer and Daubechies. In particular, the authors show that wavelets not only provide orthonormal bases for \(L^2({\mathbb{R}})\) but can be used to characterize several other function spaces, as Lebesgue, Hardy, Sobolev, Besov and Triebel-Lizorkin spaces.

The book is concentrated on the one-dimensional theory. It succeeds to give a unified and very elegant representation of construction and properties of wavelets, based on Fourier transform. Great importance is attached to mathematically exact and complete proofs.

The reader should be familar with the language of measure theory and Fourier transform and its basic properties.

The first four chapters and Chapter 7 are devoted to the construction and characterization of wavelets. Chapter 1 deals with local sine and cosine bases of Coifman and Meyer. These bases are used to construct the wavelets of Lemarié and Meyer. In the second chapter, the commonly used method for construction of wavelets is introduced, the multiresolution analysis. This method is applied to obtain the well-known Daubechies wavelets. The next two chapters are concerned with two further classes of wavelets, the “band-limited” wavelets and the spline wavelets. Finally, Chapter 7 summarizes the wavelet theory and gives a characterization of all wavelets described before. Moreover, wavelets arising from a multiresolution analysis can exactly be characterized.

Chapters 5 and 6 are devoted to the study of different function spaces with the new wavelet tool. The notion of unconditional basis in Banach spaces is introduced, and several function spaces are characterized with appropriate wavelet bases. This treatment uses properties of Calderón-Zygmund operators.

In Chapter 8, the wavelet theory is extended to frames. Finally, in the last chapter, the algorithmic ideas of fast Fourier transform and fast wavelet transform are summarized.

For people who love precise and elegant mathematics, it will be a pleasure to read this book.

The book is concentrated on the one-dimensional theory. It succeeds to give a unified and very elegant representation of construction and properties of wavelets, based on Fourier transform. Great importance is attached to mathematically exact and complete proofs.

The reader should be familar with the language of measure theory and Fourier transform and its basic properties.

The first four chapters and Chapter 7 are devoted to the construction and characterization of wavelets. Chapter 1 deals with local sine and cosine bases of Coifman and Meyer. These bases are used to construct the wavelets of Lemarié and Meyer. In the second chapter, the commonly used method for construction of wavelets is introduced, the multiresolution analysis. This method is applied to obtain the well-known Daubechies wavelets. The next two chapters are concerned with two further classes of wavelets, the “band-limited” wavelets and the spline wavelets. Finally, Chapter 7 summarizes the wavelet theory and gives a characterization of all wavelets described before. Moreover, wavelets arising from a multiresolution analysis can exactly be characterized.

Chapters 5 and 6 are devoted to the study of different function spaces with the new wavelet tool. The notion of unconditional basis in Banach spaces is introduced, and several function spaces are characterized with appropriate wavelet bases. This treatment uses properties of Calderón-Zygmund operators.

In Chapter 8, the wavelet theory is extended to frames. Finally, in the last chapter, the algorithmic ideas of fast Fourier transform and fast wavelet transform are summarized.

For people who love precise and elegant mathematics, it will be a pleasure to read this book.

Reviewer: G.Plonka (Rostock)

##### 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 |