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On the structure of Gabor and super Gabor spaces

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Abstract

We study the structure of Gabor and super Gabor spaces inside \({L^{2}(\mathbb{R}^{2d})}\) and specialize the results to the case where the spaces are generated by vectors of Hermite functions. We then construct an isometric isomorphism between such spaces and Fock spaces of polyanalytic functions and use it in order to obtain structure theorems and orthogonal projections for both spaces at once, including explicit formulas for the reproducing kernels. In particular we recover a structure result obtained by N. Vasilevski using complex analysis and special functions. In contrast, our methods use only time-frequency analysis, exploring a link between time-frequency analysis and the theory of polyanalytic functions, provided by the polyanalytic part of the Gabor transform with a Hermite window, the polyanalytic Bargmann transform.

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Authors and Affiliations

  1. Department of Mathematics of University of Coimbra, School of Science and Technology (FCTUC), CMUC, 3001-454, Coimbra, Portugal

    Luís Daniel Abreu

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  1. Luís Daniel Abreu
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Correspondence to Luís Daniel Abreu.

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Communicated by K. Gröchenig.

This research was partially supported by CMUC/FCT and FCT post-doctoral grant SFRH/BPD/26078/2005, POCI 2010 and FSE.

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Abreu, L.D. On the structure of Gabor and super Gabor spaces. Monatsh Math 161, 237–253 (2010). https://doi.org/10.1007/s00605-009-0177-0

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  • DOI: https://doi.org/10.1007/s00605-009-0177-0

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