Title	Gabor Analysis and Algorithms [electronic resource] : Theory and Applications / edited by Hans G. Feichtinger, Thomas Strohmer
Imprint	Boston, MA : Birkhรคuser Boston : Imprint: Birkhรคuser, 1998
Connect to	http://dx.doi.org/10.1007/978-1-4612-2016-9
Descript	XVI, 496 p. online resource

In his paper Theory of Communication [Gab46], D. Gabor proposed the use of a family of functions obtained from one Gaussian by time-and frequencyยญ 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 coeffiยญ 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 seriยญ 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 instaยญ 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

1 The duality condition for Weyl-Heisenberg frames -- 2 Gabor systems and the Balian-Low Theorem -- 3 A Banach space of test functions for Gabor analysis -- 4 Pseudodifferential operators, Gabor frames, and local trigonometric bases -- 5 Perturbation of frames and applications to Gabor frames -- 6 Aspects of Gabor analysis on locally compact abelian groups -- 7 Quantization of TF lattice-invariant operators on elementary LCA groups -- 8 Numerical algorithms for discrete Gabor expansions -- 9 Oversampled modulated filter banks -- 10 Adaptation of Weyl-Heisenberg frames to underspread environments -- 11 Gabor representation and signal detection -- 12 Multi-window Gabor schemes in signal and image representations -- 13 Gabor kernels for affine-invariant object recognition -- 14 Gaborโ{128}{153}s signal expansion in optics