  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.
