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Discrete Fourier Analysis

M. W. Wong
Publication Date: 
Number of Pages: 
Pseudo-Differential Operators Theory and Applications 5
[Reviewed by
Mehdi Hassani
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Fourier analysis is a topic that is used by a wide range of people with hugely varying points of view. A notable application is in electrical engineering, including “signal analysis”, “wavelets” and “filtering.” These are some of the concepts on which the book under review focuses, maintaining, as the title indicates, an emphasis on the discrete version of the theory.

The book consists of 23 friendly and concise chapters. We may divide the book into two main parts: the first part, chapters 1–13, is motivated by signal analysis. It discusses topics related to discrete time frequency analysis and discrete wavelet analysis. The second part, chapters 14–23, starts by reviewing the concept of a Hilbert space and takes an operational point of view.

Given the large number of books about Fourier analysis, it is not an easy task to find new and interesting material to include. The author of this book is successful, in my opinion, because of the point of view he brings to the subject. We have in hand a friendly volume that will be useful for people looking to apply Fourier analysis in electrical engineering and to pseudo-differential operators, as well as for applied mathematics students and researchers.

The fluent text of the book and the exercises at the end of each chapter make the book a good textbook for advanced students. Moreover, students and instructors may also use this book as a supplement to a course in Fourier analysis.

Mehdi Hassani is a faculty member at the Department of Mathematics, Zanjan University, Iran. His main field of interest is Elementary, Analytic and Probabilistic Number Theory.  

Preface.- The Finite Fourier Transform.- Translation-Invariant Linear Operators.- Circulant Matrices.- Convolution Operators.- Fourier Multipliers.- Eigenvalues and Eigenfunctions.- The Fast Fourier Transform.- Time-Frequency Analysis.- Time-Frequency Localized Bases.- Wavelet Transforms and Filter Banks.- Haar Wavelets.- Daubechies Wavelets.- The Trace.- Hilbert Spaces.- Bounded Linear Operators.- Self-Adjoint Operators.- Compact Operators.- The Spectral Theorem.- Schatten–von Neumann Classes.- Fourier Series.- Fourier Multipliers on S1.- Pseudo-Differential Operators on S1.- Pseudo-Differential Operators on Z.- Bibliography.- Index.