# Wavelets. A Student Guide

There are many ways in which wavelets can be introduced, depending on the mathematical knowledge of the student's or readers' background: one may take a linear algebra approach, a signal processing approach, an approach starting from approximation theory, in particular splines, or Fourier analysis, or a general analysis approach, or even use a background from theoretical physics. And there is the digital algorithmic approach versus the continuous analytic vision. There are almost as many ways to introduce wavelets as there are books written with that purpose. The present one, is one I like very much if it is to be used to bring mathematics students at the level of their first or second year at the university into contact with wavelets. If I had to teach such a course, this would definitely be my first choice.

Not only does it bring the subject in a most suitable and systematic way that, I am sure, mathematics students are used to and probably appreciate most. It is also following some good rules of didactics taking the students by the hand and bringing them to a higher level of understanding, ensuring that at least the bulk of the students does not declutch. A lot of effort is put into taking the rungs of the ladder at just the right pace, not boringly slow or not frighteningly fast, and always placing a chapter in the proper context: what has been achieved, and where do we want to go?

Faithful to these principles, the first chapter is just a survey, summarizing the content of the whole book, even introducing the Haar wavelet, which being the simplest possible wavelet, does not require much analysis, but it does illustrate the idea. The rest of the first half of the text does not really deal with wavelets at all but introduces vector spaces, inner products, projections, etc., first in $\mathbb{R}^N$, but once this has been explained, all the concepts are shown to be not much different when it is generalized to sequences $\ell^2$ or square integrable functions that form $L^2$. Of course the latter are Hilbert spaces which requires some more advanced elements such as convergence, measure theory, integration, etc. It is not an in-depth analysis of all these topics, but just what is needed to move on is introduced. For example there are some considerations about a basis, density, and orthogonalization in an infinite dimensional space, but the concept of a frame is not introduced. Most of the proofs are included, some parts and some proofs are given as exercises, but again the most difficult ones are left out.

The second half of the book then treats the wavelets. First the Haar wavelet is revised for which all the wavelet concepts are introduced such as a multiresolution analysis and all its properties, the scaling and the wavelet function, the scaling (or dilation) equation. The next step is to lift this to the more general situation of a general wavelet (assuming it has a finite support), the vanishing moments and the smoothness of the wavelet and the orthogonality properties and how all these properties can be formulated as conditions on the coefficients of the scaling equation. In the next chapter all this is made more concrete by deriving, drawing, and analysing the Daubechies wavelets for $N=2$ (for $N>2$ and other families the analysis is much shorter). In a last chapter, the Fourier-domain treatment of all this is discussed. Fourier analysis is again only introduced at a level just sufficient to do the computations, which avoids the deeper analysis requiring the massive body of Lebesgue integration and the subtleties of Fourier analysis.

This survey illustrates the level of the approach and also the content is purely mathematical, avoiding algorithms, applications, linear algebra, etc. Each chapter is concluded with a long list of exercises (there are about 230 exercises in the whole text). They respect the level of the text and are not trivial nor exceptionally demanding. A remarkable feature of the book is the use of something like ▶ earmarking many sections typeset in a slightly smaller font. They give some extra information or warning, not really essential to follow the flow of the exposition. It is as if the authors whisper some extra information into the ear of the reader while he/she is studying the text. The authors give also several suggestions in the introduction on how a selection can be made from the text to cover a shorter course, and in an appendix they discuss pointers to the literature and they do this chapter by chapter and in particular also for the exercises. I think this is a book perfect for what it is intended to be and it is obviously prepared with great care for precision, level of complication, and it has very good didactical qualities.

**Submitted by Adhemar Bultheel |

**4 / Mar / 2017