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Timefrequency 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 timefrequency shifts) for the anal ysis of functions and operators. Timefrequency 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 timefrequency analysis, whence the title of the book. The topics range from the elemen tary theory of the shorttime Fourier transform and classical results about the Wigner distribution via the recent theory of Gabor frames to quantita tive methods in timefrequency 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 timefrequency analysis. Timefrequency 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.
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