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

Karlheinz Gröchenig (Author)

Book Description

ISBN-10: 0817640223 | ISBN-13: 978-0817640224 | Publication Date: December 15, 2000 | Edition: 1

This work presents a systematic introduction to time-frequency analysis understood as a central area of applied harmonic analysis, while at the same time honouring its interdisciplinary origins. Important principles include: classic Fourier analysis as a tool that is central in modern mathematics; the mathematical structures based on the operations of translation and modulations (i.e: the Heisenberg group); the many forms of the uncertainty principle; and the omnipresence of Gaussian functions, both in the methodology of proofs and in important statements.

Editorial Reviews

Review

"Foundations of Time-Frequency Analysis provides a clear and thorough exposition of some of the fundamental results in the theory and gives some important perspectives on a rapidly growing field . . . An important feature of the book is complete, detailed proofs of all claims and extensive motivation of topics . . . The author has chosen topics that illuminate a path toward some of the most interesting and challenging research areas in mathematical time-frequency analysis. A graduate student or researcher seeking research problems in this area can come to no better source . . . The book is definitely suitable for a graduate-level course in mathematical time-frequency analysis. It assumes a background in real analysis, Fourier analysis and Hilbert spaces. It is also suitable for self-study, as the exposition is superb."   —Mathematical Reviews "This book is written by one of the leading experts in Gabor analysis and deserves considerable interest. It gives a unified approach to most of the modern theory for time-frequency analysis from a mathematician's point of view, with new proofs of many recent results."   —Zentralblatt Math "In contrast with the crowded market for wavelet expositions, this book has no up-to-date competitors in its niche, the rigorous mathematical theory. Groechenig makes contact with representation theory, operator algebra theory, and concludes with applications to pseudodifferential operators. But elsewhere he develops inequalities with implications for numerical analysis that should interest signal-processing engineers. He also explains how foundational investigations of quantum theory provided the subject with some of its original impetus.... Throughout, the author displays generosity to his readers as well as fastidious attention to mathematical detail. Of potential interest to graduate students and faculty in mathematics, physics, electrical engineering, and computer science."   —Choice

Product Details

• Hardcover: 374 pages
• Publisher: Birkhäuser Boston; 1 edition (December 15, 2000)
• Language: English
• ISBN-10: 0817640223
• ISBN-13: 978-0817640224
• Product Dimensions: 9.3 x 6.2 x 1.1 inches
• Shipping Weight: 1.1 pounds (View shipping rates and policies)
• Average Customer Review: 5.0 out of 5 stars  See all reviews (2 customer reviews)
• Amazon Best Sellers Rank: #844,202 in Books (See Top 100 in Books)

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Most Helpful Customer Reviews

11 of 11 people found the following review helpful:

5.0 out of 5 stars long awaited mathematical introduction to time-frequency, March 20, 2001

By 

"hgfei" (Vienna (AUSTRIA)) - See all my reviews

This review is from: Foundations of Time-Frequency Analysis (Applied and Numerical Harmonic Analysis) (Hardcover)

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.

4 of 4 people found the following review helpful:

5.0 out of 5 stars An Immediate Classic, February 13, 2003

By 

Peter G. Casazza (Columbia, MO) - See all my reviews

This review is from: Foundations of Time-Frequency Analysis (Applied and Numerical Harmonic Analysis) (Hardcover)

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.