On planar sampling with Gaussian kernel in spaces of bandlimited functions

TL;DR

This paper characterizes the conditions under which bandlimited signals can be stably reconstructed from samples obtained by convolving with Gaussian kernels at discrete points in the plane.

Contribution

It provides a complete description of discrete sampling sets allowing stable reconstruction from Gaussian-smoothed samples of bandlimited functions.

Findings

Identifies conditions for stable sampling with Gaussian kernels.

Characterizes discrete sets enabling reconstruction.

Advances understanding of sampling in Gaussian kernel spaces.

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 $\{f \ast G_{\alpha} (\lambda)\}_{\lambda \in \Lambda}$.