##
**Wavelet transforms and their applications.**
*(English)*
Zbl 1019.94003

Boston, MA: Birkhäuser. xv, 565 p. (2002).

The book under review can be considered as a textbook for upper level undergraduate courses on mathematical methods in signal processing. It provides the reader with extensive preliminaries, mostly simple proofs, some historical background, and a number of worked examples and exercises. The topics range from the standard Fourier transform, through time-frequency analysis to wavelets.

The book consists of nine chapters which are organized as follows: Chapter 1 offers a short introduction to the subjects covered in the text, together with some historical information and comments.

Chapter 2 contains an overview of the theory of Hilbert spaces that is necessary for a good understanding of the mathematical concepts that this book deals with: time-frequency and time-scale analysis. The chapter starts with basic definitions and examples, and quickly proceeds to the theory of linear operators on Hilbert spaces. Special emphasis is given to the notion of an orthonormal basis of a Hilbert space. This chapter ends with 95 problems.

Chapter 3 is another of the introductory chapters. This one deals with the theory of Fourier transforms and their applications in the study of ordinary and partial differential equations. It also contains a short section on the use of the Fourier transform in probability theory. Other topics include the Heisenberg uncertainty principle and the Shannon sampling theorem. This last subject, however, might leave the reader interested in applications of these mathematical methods to engineering hungry for more.

Chapter 4 is devoted to the study of time-frequency signal analysis. It introduces the continuous Gabor transform (also known as the windowed Fourier transform) together with its fundamental properties. Gabor systems are presented as a discrete sampling of the continuous transformation. These discrete systems possess some intriguing properties, like, e.g., the Balian-Low phenomenon, which leads, in turn, to the notions of oversampling and undersampling.

Chapter 5 continues with the study of time-frequency analysis and introduces the Wigner-Ville distribution (also known as the Wigner transform), a tool that has not made it yet to standard lower level texts.

Chapter 6 is where the wavelets and continuous wavelet transforms are introduced. Examples include Haar, Shannon, and Morlet wavelets. The important notions of a frame and a frame operator are also included in this chapter.

Chapter 7 is a continuation of the discussion of the time-scale analysis and focuses on the study of the multiresolution analysis and its applications to the construction of wavelets. These constructions include spline wavelets (like Franklin or Battle-Lemarié wavelets) and the compactly supported orthonormal wavelets of Daubechies. The chapter ends with a short section on discrete wavelet transforms, as introduced by Mallat (the pyramid algorithm). The topics in this chapter are described from a theoretical point of view and they do not address the issues of implementation.

Chapter 8 introduces a notion of harmonic wavelets, which are a modification of the wavelet construction through a version of a multiresolution analysis.

Chapter 9 deals with the applications of wavelet transforms in the analysis of turbulence. This is the only chapter truly oriented towards applications, however it is this reviewer’s belief that the material presented in this chapter is too difficult for an average undergraduate student, which stands in contrast with the presentation of the previous eight chapters.

The book consists of nine chapters which are organized as follows: Chapter 1 offers a short introduction to the subjects covered in the text, together with some historical information and comments.

Chapter 2 contains an overview of the theory of Hilbert spaces that is necessary for a good understanding of the mathematical concepts that this book deals with: time-frequency and time-scale analysis. The chapter starts with basic definitions and examples, and quickly proceeds to the theory of linear operators on Hilbert spaces. Special emphasis is given to the notion of an orthonormal basis of a Hilbert space. This chapter ends with 95 problems.

Chapter 3 is another of the introductory chapters. This one deals with the theory of Fourier transforms and their applications in the study of ordinary and partial differential equations. It also contains a short section on the use of the Fourier transform in probability theory. Other topics include the Heisenberg uncertainty principle and the Shannon sampling theorem. This last subject, however, might leave the reader interested in applications of these mathematical methods to engineering hungry for more.

Chapter 4 is devoted to the study of time-frequency signal analysis. It introduces the continuous Gabor transform (also known as the windowed Fourier transform) together with its fundamental properties. Gabor systems are presented as a discrete sampling of the continuous transformation. These discrete systems possess some intriguing properties, like, e.g., the Balian-Low phenomenon, which leads, in turn, to the notions of oversampling and undersampling.

Chapter 5 continues with the study of time-frequency analysis and introduces the Wigner-Ville distribution (also known as the Wigner transform), a tool that has not made it yet to standard lower level texts.

Chapter 6 is where the wavelets and continuous wavelet transforms are introduced. Examples include Haar, Shannon, and Morlet wavelets. The important notions of a frame and a frame operator are also included in this chapter.

Chapter 7 is a continuation of the discussion of the time-scale analysis and focuses on the study of the multiresolution analysis and its applications to the construction of wavelets. These constructions include spline wavelets (like Franklin or Battle-Lemarié wavelets) and the compactly supported orthonormal wavelets of Daubechies. The chapter ends with a short section on discrete wavelet transforms, as introduced by Mallat (the pyramid algorithm). The topics in this chapter are described from a theoretical point of view and they do not address the issues of implementation.

Chapter 8 introduces a notion of harmonic wavelets, which are a modification of the wavelet construction through a version of a multiresolution analysis.

Chapter 9 deals with the applications of wavelet transforms in the analysis of turbulence. This is the only chapter truly oriented towards applications, however it is this reviewer’s belief that the material presented in this chapter is too difficult for an average undergraduate student, which stands in contrast with the presentation of the previous eight chapters.

Reviewer: Wojciech Czaja (College Park)

### MSC:

94A12 | Signal theory (characterization, reconstruction, filtering, etc.) |

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

65T60 | Numerical methods for wavelets |