On Planar Sampling with Gaussian Kernel in Spaces of Bandlimited Functions

Abstract

Let $I=(a,b)\times (c,d)\subset {\mathbb R}_{+}^2$ be an index set and let $\{G_{\alpha }(x) \}_{\alpha \in I}$ be a collection of Gaussian functions, i.e. $G_{\alpha }(x) = \exp (-\alpha _1 x_1^2 - \alpha _2 x_2^2)$, where $\alpha = (\alpha _1, \alpha _2) \in I, \, x = (x_1, x_2) \in {\mathbb R}^2$. We present a complete description of the uniformly discrete sets $\Lambda \subset {\mathbb R}^2$ such that every bandlimited signal f admits a stable reconstruction from the samples $\{f *G_{\alpha } (\lambda )\}_{\lambda \in \Lambda }$.