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.

翻译：本文研究了由卷积核定义的卷积算子驱动的$\ell^2(\mathbb{Z})$上的动态采样问题。该问题源于从时空样本中恢复带限热扩散场及其离散模拟的需求。在本章节中，我们回顾了有限维情形下的近期结果，并将这些发现推广到无限维情形，重点研究了时空采样集的密度特性。