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.

Zusammenfassung
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] .

Inhalt
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 Gabor's signal expansion in optics.