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 Gaborโs signal expansion in optics

1. Mathematics
2. Functional analysis
3. Applied mathematics
4. Engineering mathematics
5. Mathematics
6. Applications of Mathematics
7. Signal
8. Image and Speech Processing
9. Appl.Mathematics/Computational Methods of Engineering
10. Functional Analysis