Foundations of Time-Frequency Analysis

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 book presents the first systematic introduction to time-frequency analysis understood as a central area of applied harmonic analysis, while at the same time honoring 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. Topics and Features: * Underlying thread throughout the book is the idea of a joint time-frequency representation and its conflict with the uncertainty principle * Unified and systematic introduction of the mathematical foundations of time-frequency analysis on the basis of classical harmonic analysis to obtain core results. * Emphasis of the interdisciplinary aspects of the subject and its connections to other disciplines within and outside mathematics. * new results in the modern theory of Gabor frames and the quantitative measurement of time-frequency content through the theory of modulation spaces * the role of pseudodifferential operators in time-frequency analysis. Mathematicians, physicists and engineers in signal and image analysis, will find an authoritative, systematic introduction to this active field of modern analysis and applications. Researchers and professionals in wavelets and mathematical signal analysis will also find the book a useful resource.