Gröchenig K. Foundations of Time-Frequency Analysis

Birkhäuser, 2001. — 367 p.Time-frequency analysis is a modern branch of harmonic analysis. It comprises all those parts of mathematics and its applications that use the structure of translations and modulations (or time-frequency shifts) for the analysis of functions and operators. Time-frequency analysis is a form of local Fourier analysis that treats time and frequency simultaneously and symmetrically.
The topics range from the elementary theory of the short-time Fourier transform and classical results about the Wigner distribution via the recent theory of Gabor frames to quantitative 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 orientation is toward the detailed mathematical investigation of the rich and elegant structures underlying time-frequency analysis.
This book is written by a mathematician. This means that it is written in the language and style of mathematics and with its conventions and standards of rigor. As a result the book will appeal first to mathematicians, but it is also accessible to engineers and physicists who have a more theoretical orientation. The book is written on an introductory level so that it will be accessible to a graduate student in the second year. The main prerequisites are a solid course in analysis and some Hilbert space theory. Only in a few places is more background required. On the other hand, this book will also appeal to the experts because I have reworked and incorporated a large amount of research that appears here in book form for the first time.
To help the reader I have tried to carry out most arguments in full detail and to minimize the use of phrases like "It is easy to see" or "The details are left to the reader." Therefore I have often chosen to execute a tedious, ugly, or long calculation when it would have been simpler and more convenient to say, "After an easy calculation we obtain . " In case of doubt I have chosen redundancy over brevity. I hope that these stylistic choices help the beginner to enter this beautiful area of time-frequency analysis. As a principle of writing I have tried to reinvent the subject andBasic Fourier Analysis
Time-Frequency Analysis and the Uncertainty Principle
The Short-Time Fourier Transform
Quadratic Time-Frequency Representations
Discrete Time-Frequency Representations: Gabor Frames
Existence of Gabor Frames
The Structure of Gabor Systems
Zak Transform Methods
The Heisenberg Group: A Different Point of View
Wavelet Transforms
Modulation Spaces
Gabor Analysis of Modulation Spaces
Window Design and Wiener's Lemma
Pseudodifferential Operators