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

TL;DR

This paper investigates sampling and reconstruction in shift-invariant spaces generated by bivariate Gaussian functions, identifying stable sampling patterns and thresholds, with applications to Gabor frames.

Contribution

It characterizes stable sampling patterns near critical density for bivariate Gaussian shifts and explores the impact of line slopes, providing new Gabor frame examples.

Findings

Stable reconstruction for certain lattice and line patterns near Landau's density limit

Reconstruction fails at the critical density for some sampling patterns

New Gabor frames with non-standard lattices close to volume 1

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.