Foundations of Time-Frequency Analysis (Applied and Numerical Harmonic Analysis) 2001st Edition by Karlheinz Gröchenig (PDF)

Description

Time-frequency analysis is a modern branch of harmonic analysis. It com­ prises all those parts of mathematics and its applications that use the struc­ ture of translations and modulations (or time-frequency shifts) for the anal­ ysis of functions and operators. Time-frequency analysis is a form of local Fourier analysis that treats time and frequency simultaneously and sym­ metrically. My goal is a systematic exposition of the foundations of time-frequency analysis, whence the title of the book. The topics range from the elemen­ tary theory of the short-time Fourier transform and classical results about the Wigner distribution via the recent theory of Gabor frames to quantita­ tive methods in time-frequency analysis and the theory of pseudodifferential operators. This book is motivated by applications in signal analysis and quantum mechanics, but it is not about these applications. The main ori­ entation is toward the detailed mathematical investigation of the rich and elegant structures underlying time-frequency analysis. Time-frequency analysis originates in the early development of quantum mechanics by H. Weyl, E. Wigner, and J. von Neumann around 1930, and in the theoretical foundation of information theory and signal analysis by D.

User’s Reviews

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

Reviews from Amazon users which were colected at the time this book was published on the website:

⭐This book is an essential tool for anyone working intime-frequency analysis. It is so important that I boughtTWO copies – one for home and one for work. Hardlya day goes by that I do not look into this book for somegem which clearly explains a difficult topic. This is anarea with a tremendous amount of literature. Some of itis written in such generality and obscure notation that onecannot recognize even a familiar theorem in some of these papers.This book contains all the best and clearest proofs of theresults. It also organizes this subject exceptionally well sothat it all finally “flows” the way it should. I could discussthe book section by section to point out the importantpoints. But this review would then be as long as the book.All I can say is that no one should even dream of working intime-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 theunderstanding and enjoyment of my professional life.

⭐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.

⭐The best source of Time-Frequency Analysis topics, which explains the subject in a clear and clever way, with lots of references as well.

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