Convolutional dynamical sampling and some new results

TL;DR

This paper investigates the dynamical sampling problem on infinite-dimensional spaces driven by convolution operators, extending finite-dimensional results to analyze the density of sampling sets for heat diffusion fields.

Contribution

It extends recent finite-dimensional dynamical sampling results to infinite-dimensional settings involving convolution operators, focusing on sampling set density.

Findings

Extended finite-dimensional results to infinite-dimensional cases.

Analyzed the density of space-time sampling sets for convolution-driven processes.

Provided new insights into sampling theory for heat diffusion fields.

Abstract

In this work, we explore the dynamical sampling problem on $\ell^2(\mathbb{Z})$ driven by a convolution operator defined by a convolution kernel. This problem is inspired by the need to recover a bandlimited heat diffusion field from space-time samples and its discrete analogue. In this book chapter, we review recent results in the finite-dimensional case and extend these findings to the infinite-dimensional case, focusing on the study of the density of space-time sampling sets.