Sampling in the shift-invariant space generated by the bivariate Gaussian function

Jose Luis Romero, Alexander Ulanovskii, Ilia Zlotnikov

Publications: Working paperPreprint

Abstract

We study the space spanned by the integer shifts of a bivariate Gaussian function and the problem of reconstructing any function in that space from samples scattered across the plane. We identify a large class of lattices, or more generally semi-regular sampling patterns spread along parallel lines, that lead to stable reconstruction while having densities close to the critical value given by Landau's limit. At the critical density, we construct examples of sampling patterns for which reconstruction fails.

In the same vein, we also investigate continuous sampling along non-uniformly scattered families of parallel lines and identify the threshold density of line configurations at which reconstruction is possible. In a remarkable contrast with Paley-Wiener spaces, the results are completely different for lines with rational or irrational slopes.

Finally, we apply the sampling results to Gabor systems with bivariate Gaussian windows. As a main contribution, we provide a large list of new examples of Gabor frames with non-complex lattices having volume close to 1.
Original languageEnglish
Number of pages25
DOIs
Publication statusPublished - 23 Jun 2023

Austrian Fields of Science 2012

  • 101002 Analysis

Keywords

  • Sampling
  • Shift-invariant space
  • Gabor Frame
  • bivariate Gaussian

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