Wavelets, approximation, and statistical applications.

*(English)*Zbl 0899.62002
Lecture Notes in Statistics (Springer). 129. Berlin: Springer. xviii, 265 p. (1998).

The mathematical theory of wavelets (w.) was developed by Y. Meyer about 10 years ago. It was designed for approximation of irregular functions and was successfully applied in data compression, turbulence analysis, image and signal processing. Recently w. theory appeared to be a powerful framework for nonparametric statistical problems.

This book is a clearly written textbook. It provides an introduction to w. theory, discusses approximations and gives a lot of statistical applications. A pure scientist will find here in a condensed form the results on the construction of w. that are scattered around in the research literature. A practical person will be able to use w. via the available software codes.

The simplest w. basis, the Haar basis, is presented in Chapter 2. The basic idea of space/frequency multiresolution is given in Chapter 3. The basics of w. theory and the actual construction of w. are presented in Chapters 5 and 6. In Chapter 7 the Daubechies compactly supported w. are built. Chapters 8 and 9 study the approximation properties of w. decomposition and give an introduction to Besov spaces which are closely related with w. expansions. Chapter 10 is devoted to linear w. density estimators and to soft and hard thresholding of w. coefficients, which leads to nonlinear estimating procedures. Risk bounds and the optimal rate of convergence of the estimators are discussed. The nonlinearity guarantees smoothness adaptivity of the estimators, as it is shown in Chapter 11. The final Chapter 12 discusses computational aspects and an interactive software interface. In the appendix the address for the XploRe software sources is given.

Together with the authors I hope that the book will help to construct bridges between the different groups of scientists working in either pure or applied mathematics.

This book is a clearly written textbook. It provides an introduction to w. theory, discusses approximations and gives a lot of statistical applications. A pure scientist will find here in a condensed form the results on the construction of w. that are scattered around in the research literature. A practical person will be able to use w. via the available software codes.

The simplest w. basis, the Haar basis, is presented in Chapter 2. The basic idea of space/frequency multiresolution is given in Chapter 3. The basics of w. theory and the actual construction of w. are presented in Chapters 5 and 6. In Chapter 7 the Daubechies compactly supported w. are built. Chapters 8 and 9 study the approximation properties of w. decomposition and give an introduction to Besov spaces which are closely related with w. expansions. Chapter 10 is devoted to linear w. density estimators and to soft and hard thresholding of w. coefficients, which leads to nonlinear estimating procedures. Risk bounds and the optimal rate of convergence of the estimators are discussed. The nonlinearity guarantees smoothness adaptivity of the estimators, as it is shown in Chapter 11. The final Chapter 12 discusses computational aspects and an interactive software interface. In the appendix the address for the XploRe software sources is given.

Together with the authors I hope that the book will help to construct bridges between the different groups of scientists working in either pure or applied mathematics.

Reviewer: Oleksandr Kukush (Kyiv)

##### MSC:

62-02 | Research exposition (monographs, survey articles) pertaining to statistics |

62G07 | Density estimation |

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