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Feichtinger, Hans G.

Gabor expansions of signals: computational aspects and open questions. (English) Zbl 1426.42030

Boggiatto, Paolo (ed.) et al., Landscapes of time-frequency analysis. Based on talks given at the inaugural conference on aspects of time-frequency analysis, Turin, Italy, July 5–7, 2018. Cham: Birkhäuser. Appl. Numer. Harmon. Anal., 173-206 (2019).
From the author’s summary: “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 spaces, in particular modulation spaces …In contrast, the applied literature gives the impression that the computation of dual Gabor windows in the standard situation …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 the numerical realization of numerical concepts. It has been in many cases the inspiration for the derivation of theoretical results …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.”
This survey has 115 bibliographical references and is divided into 13 sections titled: Introduction, Goals for numerical Gabor analysis, The standard literature, The classical frame algorithm, The idea of double preconditioning, The Janssen representation, Gabor multipliers, Numerical illustrations, Computations benefitting from theory, What are good Gabor systems?, The role of the Zak transform, Transfer from \(R\) to \(Z_n\), and Acknowledgements.
The paper ends with a suggestion that in order to avoid duplication for people working on the issues raised in this work, they should contact the author to coordinate their efforts.
For the entire collection see [Zbl 1411.35009].
Reviewer: Richard A. Zalik (Auburn)

Cited in 4 Documents

MSC:

42C40 Nontrigonometric harmonic analysis involving wavelets and other special systems
42C15 General harmonic expansions, frames
65T60 Numerical methods for wavelets
94A12 Signal theory (characterization, reconstruction, filtering, etc.)

Keywords:

numerical Gabor analysis; frame algorithm; double preconditioning; Janssen representation; Zak transform
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References:

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