# Wavelet Transforms and Their Applications (2nd ed.)

The original version of 2002 has been thoroughly revised and recent evolutions in the area of wavelet research have been added. The original concept of the book is maintained, i.e., it can be seen as a reference text or as a study book, complete with definitions, theorems, proofs and exercises. There is mathematical rigor, yet abstraction is only allowed when justified.

The title may suggest that this book is only about wavelets and all the successful applications. However, the 'introduction' is very throughout and after some historical survey, you will find an extensive discussion of Fourier analysis (Fourier transform, Fourier series, DFT, FFT,...) and Hilbert spaces. Of course the latter are important to introduce (bi)orthogonal bases, frames, etc., which are important in wavelet analysis.

Time-frequency analysis comes into the picture with Gabor (both continuous and discrete) and Zak transforms and the Wigner-Ville and the ambiguity distributions (again continuous as well as discrete transforms are analysed). So it is only on page 337 that the wavelets as such show up. The continuous (CWT) and discrete (DWT) wavelet transforms in chapter 6 and multiresolution analysis (MRA) in chapter 7 form the core wavelet chapters. The remaining chapters treat more advanced or recent topics like a p-MRA on the positive real halfline, nonuniform MRA and the Newland harmonic wavelets combining the short-time Fourier transform and the CWT.

The applications are distributed throughout. The use of Fourier analysis in the solution of ordinary and partial differential equations and integral equations is classical and embedded in the Fourier analysis chapter. Examples of the theory are included in all chapters. The closing chapter is completely devoted to Fourier and wavelet analysis of turbulence. This subject is relatively new and it is more like a survey featuring properties of Navier-Stokes equations and difficulties that one meets when solving them with Fourier analysis. That involves topics like fractals, multifractals and singularities of turbulence. Subsequently solutions were proposed based on wavelet analysis like adaptive wavelets (Farge et al.) and wavelet transformed Navier-Stokes equations (Memeveau et al.)

The exponential development and success of wavelet analysis is due to a simultaneous alertness and collaboration between mathematicians, engineers, and theoretical physicists. So there are several approaches to the topic. On avarage, the continuous transforms are the favorites of mathematicians and physicists, the engineering applications in (digital) signal and image analysis and the numerical solution of functional equations are favoring the discrete transforms. The former often emphasize the analysis aspect, the latter often find that they are better off with a computational (linear) algebraic approach. This book tries to keep a good balance between both, but I believe the authors have a reasonable bias towards the former camp of the CWT. Although the discrete transforms are all presented and given proper attention, the continuous transforms come up first to pave the way for the discrete versions. The applications are more the applications in a mathematician's vision: functional equations, analysis of turbulence (emphasis on analysis), while engineers would call subjects like for example image and signal processing and compression or the computational and numerical aspects of the equation solvers, as true applications. Those engineering kind of applications is not exactly what you should look for in this book. On the other hand, all the elements for practical (also engineering) applications are introduced. However as much as the text keeps far away from unnecessary abstraction like 'harmonic analysis on locally compact groups', it keeps away from computer programming too.

The book is an up to date reference work on univariate Fourier and wavelet analysis including recent developments in multiresolution, wavelet analysis, and applications in turbulence. The systematic construction of the chapters with extensive lists of exercises make it also very suitable for teaching.

**Submitted by Adhemar Bultheel |

**4 / Feb / 2015