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Wavelets: A Student Guide

Peter Nickolas
Cambridge University Press
Publication Date: 
Number of Pages: 
Australian Mathematical Society Lecture Series 24
[Reviewed by
Salim Salem
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In Wavelets Peter Nickolas has tried to stick to his claim that “the approach adopted here is therefore what is often referred to as elementary”. The intention is to reach a wider audience by using what any student of mathematics learns in his or her first and second years.

Nickolas divides his book into two parts and an overview. The overview (chapter 1) summarizes the book and gives an early taste of things to come. In the first part he sets the stage and recalls the results he needs to build wavelets and multiresolution analysis. In chapter 2 he defines vector spaces and gives as examples the infinite dimensional spaces \(\ell^2\) and \(L^2\). The third chapter is devoted to an in-depth study of inner product, which is a necessary step to fully understand Hilbert spaces, which are discussed in the fourth chapter.

The second part starts at chapter 5, with the most elementary example, the “Haar wavelet”. This chapter is important in two ways: first it introduces the Haar wavelet and its multiresolution analysis, and second, it shows its shortcomings, thus motivating the search for other types of wavelets, something that Nickolas takes up in the remaining chapters of his book.

This elementary approach, does not allow the author to give more than a short, theoretical and introductory account of Daubechies wavelets. But this is exactly what a newcomer to wavelets, needs. This restriction, however, keeps all applications of wavelets outside the book, which is something to regret.

What is really nice with this book is its style, which leads the student step by step through different ideas, theorems and proofs. It explains the reason behind new concepts, discusses their shortcomings, and uses these as a motivation to introduce other concepts. The book contains some graphs, which show the importance of wavelets and the speed of their convergence in approximating functions. (It is a pity that it is not compared to any other method of approximation). The exercises are a very important part of the book, and any serious student must attempt to solve all of them.

The appendix contains a list of important sources on wavelets and their construction, but does not attempt to cover all references.

To summarize: I liked the book as a short elementary introduction to a deep mathematical subject. I recommend it to any mathematics or engineering undergraduate who wish to know what wavelets are and how to construct them.

Salim Salem is Professor of Mathematics at the Saint-Joseph University of Beirut.

1. An overview
2. Vector spaces
3. Inner product spaces
4. Hilbert spaces
5. The Haar wavelet
6. Wavelets in general
7. The Daubechies wavelets
8. Wavelets in the Fourier domain
Appendix: notes on sources