Foundations of Time-Frequency Analysis (Applied and Numerical Harmonic Analysis) | Date: 30 April 2011, 06:24
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Time-Frequency Analysis is a rich source of ideas and applications in modern harmonic analysis. The history of time-frequency analysis dates back to von Neumann, Wigner, and Gabor, who considered the problems in quantum mechanics and in information theory. For many years time-frequency analysis has been pursued only in engineering, but recently, and with the development of wavelet theory, it has emerged as a thriving field of applied mathematics. This book presents the first systematic introduction to time-frequency analysis understood as a central area of applied harmonic analysis, while at the same time honoring its interdisciplinary origins. Important principles are (a) classical Fourier analysis as a tool that is central in modern mathematics, (b) the mathematical structures based on the operations of translation and modulations (i.e. the Heisenberg group), (c) the many forms of the uncertainty principle, and (d) the omnipresence of Gaussian functions, both in the methodology of proofs and in important statements. Topics and Features:
* Underlying thread throughout the book is the idea of a joint time-frequency representation and its conflict with the uncertainty principle * Unified and systematic introduction of the mathematical foundations of time-frequency analysis on the basis of classical harmonic analysis to obtain core results.
* Emphasis of the interdisciplinary aspects of the subject and its connections to other disciplines within and outside mathematics.
* new results in the modern theory of Gabor frames and the quantitative measurement of time-frequency content through the theory of modulation spaces
* the role of pseudodifferential operators in time-frequency analysis. Mathematicians, physicists and engineers in signal and image analysis, will find an authoritative, systematic introduction to this active field of modern analysis and applications. Researchers and professionals in wavelets and mathematical signal analysis will also find the book a useful resource.
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
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