Convolutional dynamical sampling and some new results
Longxiu Huang, A. Martina Neuman, Sui Tang, Yuying Xie
TL;DR
This paper investigates dynamical sampling on $\ell^2(\mathbb{Z})$ driven by a convolution operator with kernel $a\in\ell^1(\mathbb{Z})$, using space-time samples to recover the initial signal. It analyzes when the set $\Omega=\{A^{s}g: g\in\mathcal{G}, s\in\mathbb{N}_0\}$ forms a complete system or a frame for $\ell^2(\mathbb{Z})$, employing Fourier analysis and sub-lattice techniques; it derives explicit density conditions that tie sampling density to the kernel profile. A key result is that finite spatial sampling sets cannot yield frames; stability requires infinite sensor sets and, for sub-lattice schemes $\Lambda=m\mathbb{Z}+\{0,\dots,L-1\}$, necessary relations $L\ge \mathfrak{N}(t,\omega)$ and $NL\ge m$ alongside kernel-dependent density bounds. The findings bridge finite-dimensional dynamical-sampling insights with the infinite-dimensional theory, offering practical guidelines for designing stable space-time sampling schemes and outlining directions for higher-dimensional extensions and joint operator reconstruction.
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.