### Synopses & Reviews

This unified, self-contained volume provides insight into the richness of Gabor analysis and its potential for future development in applied mathematics and engineering. Mathematicians and engineers treat a range of topics covering theory as well as applications in eleven survey chapters. Taken as a whole, the work demonstrates interactions and connections among different areas in which Gabor analysis plays a critical role, including harmonic analysis, operator theory, quantum physics, numerical analysis, electrical engineering, and signal/image processing. Key features of the work: * Gives an overview of recent developments in Gabor analysis, an important tool in the understanding and use of time-frequency analysis methods in a variety of disciplines * Provides sufficient background material along with many new and previously unpublished results * Presents applications to areas such as digital and wireless communications * Up-to-date bibliographies for each chapter; useful subject index for the entire volume * Specific topics covered include: Uncertainty Principles for Time-Frequency Representations; Zak Transforms; Bracket Products for Weyl--Heisenberg Frames; Gabor Multipliers; Gabor Analysis and Operator Algebras; Integral Operators, Pseudodifferential Operators and Gabor Frames; Methods for Approximation of the Inverse (Gabor) Frame Operator; Wilson Bases Graduate students, professionals, and researchers in pure and applied mathematics, mathematical physics, electrical and communications engineering will find Advances in Gabor Analysis a comprehensive resource. Contributors: J.-P. Antoine, F. Bagarello, R. Balan, K. Bittner, H. Bölcskei, P.G. Casazza, O. Christensen, I. Daubechies, H.G. Feichtinger, J.-P. Gabardo, K. Gröchenig, D. Han, C. Heil, A.J.E.M. Janssen, M.C. Lammers, K. Nowak, T. Strohmer Also edited by Feichtinger and Strohmer---Gabor Analysis and Algorithms: Theory and Applications, ISBN 0-8176-3959-4, 1998

#### Review

"The authors have put their effort into writing articles with the necessary background to make their presentations attractive and including a complete set of references. Good introductions and detailed expositions of ideas are given in all the presentations. These features make this book an invaluable tool for researchers interested in signal analysis. The research community will benefit greatly from this publication." --Mathematical Reviews "The present book can be considered as a continuation of two previous ones: Gabor Analysis and Algorithms: Theory and Applications, H. G. Feichtinger and T. Strohmer, Eds., Birkhäuser 1998, and the book by K. Gröchenig, Foundations of Time-Frequency Analysis, Birkhäuser 2001. It contains survey chapters, but new results that have not been published previously are also included. The introductory chapter of the book, written by H. G. Feichtinger and T. Strohmer, contains a clear outline of the contents as well as some comments on the future developments in Gabor analysis. Written by leading experts in the field, the volume appeals, by its interdisciplinary character, to a large audience, both novices and experts, theoretically inclined researchers and practitioners as well. It brilliantly illustrates how application areas and pure and applied mathematics can work together with profit for all." --Revue D'Analyse Numérique et de Théorie de L'Approximation, 2004 "A mathematically sound and self-contained presentation on recent advances in the highly applicable field of Gabor analysis.... This book is a must for all interested in signal analysis, and, of course, in particular in Gabor analysis." --Monatschefte für Mathematik

#### Synopsis

The Applied and Numerical Harmonic Analysis (ANHA) book series aims to provide the engineering, mathematical, and scientific communities with significant developments in harmonic analysis, ranging from abstract har- monic analysis to basic applications. The title of the series reflects the im- portance of applications and numerical implementation, but richness and relevance of applications and implementation depend fundamentally on the structure and depth of theoretical underpinnings. Thus, from our point of view, the interleaving of theory and applications and their creative symbi- otic evolution is axiomatic. Harmonic analysis is a wellspring of ideas and applicability that has flour- ished, developed, and deepened over time within many disciplines and by means of creative cross-fertilization with diverse areas. The intricate and fundamental relationship between harmonic analysis and fields such as sig- nal processing, partial differential equations (PDEs), and image processing is reflected in our state of the art ANHA series. Our vision of modern harmonic analysis includes mathematical areas such as wavelet theory, Banach algebras, classical Fourier analysis, time- frequency analysis, and fractal geometry, as well as the diverse topics that impinge on them.

#### Synopsis

Unified, self-contained volume providing insight into the richness of Gabor analysis and its potential for development in applied mathematics and engineering. Mathematicians and engineers treat a range of topics, and cover theory and applications to areas such as digital and wireless communications. The work demonstrates interactions and connections among areas in which Gabor analysis plays a role: harmonic analysis, operator theory, quantum physics, numerical analysis, signal/image processing. For graduate students, professionals, and researchers in pure and applied mathematics, math physics, and engineering.

### Table of Contents

Foreword/H. Landau

Contributors

Introduction /H.G. Feichtinger and T. Strohmer

Uncertainty Principles for Time-Frequency Representations /K. Gröchenig

Zak Transforms with Few Zeros and the Tie /A.J.E.M Janssen

Bracket Products for Weyl--Heisenberg Frames /P.G. Casazza and M.C. Lammers

A First Survey of Gabor Multipliers /H.G. Feichtinger and K. Nowak

Aspects of Gabor Analysis and Operator Algebras /J.-P. Gabardo and D. Han

Integral Operators, Pseudodifferential Operators, and Gabor Frames /C. Heil

Methods for Approximation of the Inverse (Gabor) Frame Operator /O. Christensen and T. Strohmer

Wilson Bases on the Interval /K. Bittner

Localization Properties and Wavelet-Like Orthonormal Bases for the Lowest Landau Level /J.-P. Antoine and F. Bagarello

Optimal Stochastic Encoding and Approximation Schemes using Weyl--Heisenberg Sets /R. Balan and I. Daubechies

Orthogonal Frequency Division Multiplexing Based on Offset QAM /H. Bölcskei

Index