The book "Foundations of Time-Frequency Analysis" by Karlheinz Gröchenig is a comprehensive and systematic exposition of the mathematical foundations of time-frequency analysis. It covers a wide range of topics, from basic Fourier analysis to advanced theories such as Gabor frames, modulation spaces, and pseudodifferential operators. The author aims to provide a detailed and rigorous treatment of the subject, making it accessible to graduate students and researchers in mathematics, engineering, and physics. The book begins with fundamental concepts and gradually delves into more complex theories, including the Heisenberg group, wavelet theory, and the structure of Gabor systems. It also includes new results and proofs, such as simplifications in the theory of modulation spaces and a new boundedness result for pseudodifferential operators. The book is structured to highlight the connections between different areas of mathematics and their applications in signal processing, quantum mechanics, and other fields. Gröchenig's writing style is detailed and pedagogical, with a focus on clarity and rigor, making it a valuable resource for both beginners and experts in the field.The book "Foundations of Time-Frequency Analysis" by Karlheinz Gröchenig is a comprehensive and systematic exposition of the mathematical foundations of time-frequency analysis. It covers a wide range of topics, from basic Fourier analysis to advanced theories such as Gabor frames, modulation spaces, and pseudodifferential operators. The author aims to provide a detailed and rigorous treatment of the subject, making it accessible to graduate students and researchers in mathematics, engineering, and physics. The book begins with fundamental concepts and gradually delves into more complex theories, including the Heisenberg group, wavelet theory, and the structure of Gabor systems. It also includes new results and proofs, such as simplifications in the theory of modulation spaces and a new boundedness result for pseudodifferential operators. The book is structured to highlight the connections between different areas of mathematics and their applications in signal processing, quantum mechanics, and other fields. Gröchenig's writing style is detailed and pedagogical, with a focus on clarity and rigor, making it a valuable resource for both beginners and experts in the field.