Time-frequency analysis is a modern branch of harmonic analysis. It comยญ prises all those parts of mathematics and its applications that use the strucยญ ture of translations and modulations (or time-frequency shifts) for the analยญ ysis of functions and operators. Time-frequency analysis is a form of local Fourier analysis that treats time and frequency simultaneously and symยญ 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 elemenยญ tary theory of the short-time Fourier transform and classical results about the Wigner distribution via the recent theory of Gabor frames to quantitaยญ 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 oriยญ 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
