Hans G. Feichtinger - Böcker

This text provides an introduction to mathematicians and engineers who want to learn about the different approaches and aspects of Gabor analysis or want to apply Gabor-based techniques to tasks in signal and image processing. It should be of interest to those working in the fields of harmonic analysis, applied mathematics, numerical analysis, engineering, signal and image processing, optics and pattern recognition. Features include: self-contained introduction to basic Gabor analysis; survey of fundamental results in Gabor theory; efficient numerical algorithms; Gabor expansions in signal and image processing; and applications in pattern recognition, filter bank design and optics.

Gabor analysis constitutes a central part of time-frequency analysis. While preserving the symmetry between the time (location) domain and the frequency (wave number) domain, Gabor analysis avoids the high degree of redundancy inherent in the continuous short-time Fourier transform. The ability to resolve details of a signal (or a system impulse response) in a two-dimensional representation, whose coefficients have a very natural interpretation, is the basis for many interesting applications in electrical engineering, and signal-and image processing making Gabor analysis an important branch of applied mathematics. This work provides an overview of recent developments in Gabor analysis. Leading scientists in various disciplines related to the subject treat a range of topics from covering theory to numerics as well as applications of Gabor analysis. The presentation unfolds systematically and includes appropriate background material in each chapter, placing the work in its proper context. Graduate students, practitioners and researchers in the areas of numerical analysis, electrical engineering and applied mathematics will finds this book a current and useful resource.

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.

In his paper Theory of Communication [Gab46], D. Gabor proposed the use of a family of functions obtained from one Gaussian by time-and frequency­ shifts. Each of these is well concentrated in time and frequency; together they are meant to constitute a complete collection of building blocks into which more complicated time-depending functions can be decomposed. The application to communication proposed by Gabor was to send the coeffi­ cients of the decomposition into this family of a signal, rather than the signal itself. This remained a proposal-as far as I know there were no seri­ ous attempts to implement it for communication purposes in practice, and in fact, at the critical time-frequency density proposed originally, there is a mathematical obstruction; as was understood later, the family of shifted and modulated Gaussians spans the space of square integrable functions [BBGK71, Per71] (it even has one function to spare [BGZ75] . . . ) but it does not constitute what we now call a frame, leading to numerical insta­ bilities. The Balian-Low theorem (about which the reader can find more in some of the contributions in this book) and its extensions showed that a similar mishap occurs if the Gaussian is replaced by any other function that is "reasonably" smooth and localized. One is thus led naturally to considering a higher time-frequency density.

The chapters in this volume are based on talks given at the inaugural Aspects of Time-Frequency Analysis conference held in Turin, Italy from July 5-7, 2017, which brought together experts in harmonic analysis and its applications.

This contributed volume features chapters based on talks given at the second international conference titled Aspects of Time-Frequency Analysis (ATFA 19), held at Politecnico di Torino from June 25th to June 27th, 2019.

This contributed volume features chapters based on talks given at the second international conference titled Aspects of Time-Frequency Analysis (ATFA 19), held at Politecnico di Torino from June 25th to June 27th, 2019.

In Western Civilization Mathematics and Music have a long and interesting history in common, with several interactions, traditionally associated with the name of Pythagoras but also with a significant number of other mathematicians, like Leibniz, for instance. Mathematical models can be found for almost all levels of musical activities from composition to sound production by traditional instruments or by digital means. Modern music theory has been incorporating more and more mathematical content during the last decades. This book offers a journey into recent work relating music and mathematics. It contains a large variety of articles, covering the historical aspects, the influence of logic and mathematical thought in composition, perception and understanding of music and the computational aspects of musical sound processing. The authors illustrate the rich and deep interactions that exist between Mathematics and Music.

Pseudo-differential operators were initiated by Kohn, Nirenberg and Hörmander in the sixties of the last century. Beside applications in the general theory of partial differential equations, they have their roots also in the study of quantization first envisaged by Hermann Weyl thirty years earlier. Thanks to the understanding of the connections of wavelets with other branches of mathematical analysis, quantum physics and engineering, such operators have been used under different names as mathematical models in signal analysis since the last decade of the last century.The volume investigates the mathematics of quantization and signals in the context of pseudo-differential operators, Weyl transforms, Daubechies operators, Wick quantization and time-frequency localization operators. Applications to quantization, signal analysis and the modern theory of PDE are highlighted.