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Gabor Expansions of Signals: Computational Aspects and Open Questions

  • Hans G. FeichtingerEmail author
Chapter
Part of the Applied and Numerical Harmonic Analysis book series (ANHA)

Abstract

In the last 40 years the foundations of Gabor analysis, even in the context of locally compact Abelian (LCA) groups, have been widely developed. We know a lot about function spaces, in particular modulation spaces, characterization of these spaces via Gabor expansions, or mapping properties of operators between such spaces, even the description of solutions for PDEs can nowadays be given in this context. In contrast, the applied literature gives the impression that the computation of dual Gabor windows in the standard situation, i.e. for the Hilbert space \({{{\varvec{L}}^2}({\mathbb R})}\), and a time–frequency lattice of the form \(\varLambda = a {\mathbb Z} \times {b} {{\mathbb Z}}\) is still the most important problem in (numerical) Gabor analysis. The emphasis of this note is on the value of numerical work, which is much more than just numerical realization of theoretical concepts. It has been in many cases the inspiration for the derivation of theoretical results, based on sometimes surprising observations or systematic numerical simulations. According to our experience, numerical Gabor analysis provides a lot of additional insight about the concrete situation; it may suggest new directions and ask for new theory, but of course efficient algorithms often make use of underlying theory. Overall, we observe that there is an urgent need for a stronger link between computational and theoretical Gabor analysis. The note also contains a number of suggestions and even conjectures which are likely to encourage research in the direction indicated above.

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Authors and Affiliations

  1. 1.Faculty of Mathematics (NuHAG)University of ViennaViennaAustria

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