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Gabor Expansions of Signals: Computational Aspects and Open Questions

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Landscapes of Time-Frequency Analysis

Part of the book series: Applied and Numerical Harmonic Analysis ((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.

Summer 2017 and 2018 TU Munich, Inst. Theor. Inf. Science

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Notes

  1. 1.

    Recall that at the MATLAB command line the “\(*\)” represents matrix multiplication resp. matrix–vector multiplication!

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  1. Faculty of Mathematics (NuHAG), University of Vienna, Vienna, Austria

    Hans G. Feichtinger

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Editors and Affiliations

  1. Dipartimento di Matematica “G. Peano”, University of Torino, Turin, Italy

    Paolo Boggiatto

  2. Dipartimento di Matematica “G. Peano”, University of Torino, Turin, Italy

    Elena Cordero

  3. Faculty of Mathematics, University of Vienna, Vienna, Austria

    Maurice de Gosson

  4. Faculty of Mathematics, University of Vienna, Vienna, Austria

    Hans G. Feichtinger

  5. Department of Mathematical Sciences, Polytechnic University of Torino, Turin, Italy

    Fabio Nicola

  6. Dipartimento di Matematica “G. Peano”, University of Torino, Turin, Italy

    Alessandro Oliaro

  7. Department of Mathematical Sciences, Polytechnic University of Torino, Turin, Italy

    Anita Tabacco

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Feichtinger, H.G. (2019). Gabor Expansions of Signals: Computational Aspects and Open Questions. In: Boggiatto, P., et al. Landscapes of Time-Frequency Analysis. Applied and Numerical Harmonic Analysis. Birkhäuser, Cham. https://doi.org/10.1007/978-3-030-05210-2_7

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