Foundations of time-frequency analysis
著者
書誌事項
Foundations of time-frequency analysis
（Applied and numerical harmonic analysis / series editor, John J. Benedetto）
Birkhäuser, c2001
- : us
- : sz
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注記
Includes bibliographical references (p. 335-353) and index
内容説明・目次
- 巻冊次
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: us ISBN 9780817640224
内容説明
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.
- 巻冊次
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: sz ISBN 9783764340223
内容説明
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 title 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 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.
目次
- Basic Fourier Analysis Time-Frequency Analysis and the Uncertainty Principle The Short-Time Fourier Transform Quadratic time-frequency representations
- gabor frames
- existence of gabor frames
- the structure of gabor systems
- zak transform methods
- the heisenberg group
- wavelet transforms
- modulation spaces
- gabor analysis of modulation spaces
- window design and wiener's lemma
- pseudodifferential operators.
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