"Foundations of Time-Frequency Analysis" is a comprehensive mathematical text that systematically presents the core concepts of time-frequency analysis. The book is written by Karlheinz Gröchenig and is part of the Applied and Numerical Harmonic Analysis series. It covers a wide range of topics, including Fourier analysis, the short-time Fourier transform, the Wigner distribution, Gabor frames, and pseudodifferential operators. The book is motivated by applications in signal analysis and quantum mechanics, but its primary focus is on the mathematical foundations of time-frequency analysis. Time-frequency analysis is a modern branch of harmonic analysis that uses the structure of translations and modulations for the analysis of functions and operators. It is a form of local Fourier analysis that treats time and frequency simultaneously and symmetrically. The book provides a detailed mathematical investigation of the rich and elegant structures underlying time-frequency analysis. The book traces the origins of time-frequency analysis to the early development of quantum mechanics by H. Weyl, E. Wigner, and J. von Neumann around 1930, and to the theoretical foundation of information theory and signal analysis by D. Gabor in 1946. It also highlights the contributions of Guido Janssen, who established time-frequency analysis as an independent mathematical field around 1980. The book is written by a mathematician and is intended for a mathematical audience, but it is also accessible to engineers and physicists with a theoretical orientation. It is written on an introductory level and is suitable for graduate students in the second year. The main prerequisites are a solid course in analysis and some Hilbert space theory. The book includes a detailed introduction to the theory of modulation spaces, which are new function spaces introduced for the quantitative theory of time-frequency analysis. It also includes a number of hitherto unpublished results and generalizations. The book is written in a style that emphasizes clarity and rigor, with detailed arguments and minimal use of phrases like "It is easy to see." The book is structured into 14 chapters, covering topics from the elementary theory of the short-time Fourier transform to the modern theory of Gabor frames, and from the Heisenberg group to pseudodifferential operators. It also includes an appendix, references, and an index. The book is a comprehensive resource for researchers and students in the field of time-frequency analysis."Foundations of Time-Frequency Analysis" is a comprehensive mathematical text that systematically presents the core concepts of time-frequency analysis. The book is written by Karlheinz Gröchenig and is part of the Applied and Numerical Harmonic Analysis series. It covers a wide range of topics, including Fourier analysis, the short-time Fourier transform, the Wigner distribution, Gabor frames, and pseudodifferential operators. The book is motivated by applications in signal analysis and quantum mechanics, but its primary focus is on the mathematical foundations of time-frequency analysis. Time-frequency analysis is a modern branch of harmonic analysis that uses the structure of translations and modulations for the analysis of functions and operators. It is a form of local Fourier analysis that treats time and frequency simultaneously and symmetrically. The book provides a detailed mathematical investigation of the rich and elegant structures underlying time-frequency analysis. The book traces the origins of time-frequency analysis to the early development of quantum mechanics by H. Weyl, E. Wigner, and J. von Neumann around 1930, and to the theoretical foundation of information theory and signal analysis by D. Gabor in 1946. It also highlights the contributions of Guido Janssen, who established time-frequency analysis as an independent mathematical field around 1980. The book is written by a mathematician and is intended for a mathematical audience, but it is also accessible to engineers and physicists with a theoretical orientation. It is written on an introductory level and is suitable for graduate students in the second year. The main prerequisites are a solid course in analysis and some Hilbert space theory. The book includes a detailed introduction to the theory of modulation spaces, which are new function spaces introduced for the quantitative theory of time-frequency analysis. It also includes a number of hitherto unpublished results and generalizations. The book is written in a style that emphasizes clarity and rigor, with detailed arguments and minimal use of phrases like "It is easy to see." The book is structured into 14 chapters, covering topics from the elementary theory of the short-time Fourier transform to the modern theory of Gabor frames, and from the Heisenberg group to pseudodifferential operators. It also includes an appendix, references, and an index. The book is a comprehensive resource for researchers and students in the field of time-frequency analysis.