Customer Reviews

Foundations of Time-Frequency Analysis (Applied and Numerical Harmonic Analysis)

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This book fills a long-standing gap in the literature. While Gabor analysis is described on equal footing with wavelets in the early literature on the subject (in the late 80's) the book market on wavelets exploded, while no "serious" (from a mathematician's point of view) introduction to Gabor analysis was available. Maybe the impossibility of having orthonormal Gabor bases was the main reason for this situation (by the Balian-Low theorem). In 13 Chapters the author takes the reader from basic Fourier analysis and fundamental facts concerning the short-time Fourier transform and uncertainty to the foundations of Gabor analysis, i.e. Gabor frames, the structure of the Gabor frame operator, Zak transform methods, the role of the Heisenberg group and to functional analytic results describing the "appropriate" function spaces (so-called modulation spaces), playing here the same role as Besov spaces play with respect to wavelet expansions. Besides summarizing known results and proving things from scratch interesting applications (e.g. to pseudo- differential operators) are provided. The book should be of great interest to mathematicians (e.g. graduate students in analysis) as well as those applied scientist who want to read about the "foundations" of this field without having to consult dozends of original papers. A great book which will have its influence on the teaching of the subject in the years to come.

This book is an essential tool for anyone working in
time-frequency analysis. It is so important that I bought
TWO copies - one for home and one for work. Hardly
a day goes by that I do not look into this book for some
gem which clearly explains a difficult topic. This is an
area with a tremendous amount of literature. Some of it
is written in such generality and obscure notation that one
cannot recognize even a familiar theorem in some of these papers.
This book contains all the best and clearest proofs of the
results. It also organizes this subject exceptionally well so
that it all finally ``flows'' the way it should. I could discuss
the book section by section to point out the important
points. But this review would then be as long as the book.
All I can say is that no one should even dream of working in
time-frequency analysis without having this book at their side.
From the moment it hit the shelves it was already the ``classical''
work in the area. And it will be that for a long time to come.
I thank the author for this significant contribution to the
understanding and enjoyment of my professional life.