# Universal Spatiotemporal Sampling Sets for Discrete Spatially Invariant Evolution Systems

## TL;DR

This paper investigates how to construct minimal spatiotemporal sampling sets for discrete invariant systems, linking algebraic, spectral, and polynomial interpolation theories to ensure complete system reconstruction.

## Contribution

It provides an algebraic characterization of universal sampling sets for convolution operators with fixed eigenvalue multiplicity, advancing understanding of spatiotemporal sampling in invariant systems.

## Key findings

- The size of the sampling set must be at least the largest eigenvalue multiplicity.
- Universal sampling sets are characterized algebraically for fixed eigenvalue multiplicity.
- Connections are established between sampling, sparse signal processing, and polynomial interpolation.

## Abstract

Let $(I,+)$ be a finite abelian group and $\mathbf{A}$ be a circular convolution operator on $\ell^2(I)$. The problem under consideration is how to construct minimal $\Omega \subset I$ and $l_i$ such that $Y=\{\mathbf{e}_i, \mathbf{A}\mathbf{e}_i, \cdots, \mathbf{A}^{l_i}\mathbf{e}_i: i\in \Omega\}$ is a frame for $\ell^2(I)$, where $\{\mathbf{e}_i: i\in I\}$ is the canonical basis of $\ell^2(I)$. This problem is motivated by the spatiotemporal sampling problem in discrete spatially invariant evolution systems. We will show that the cardinality of $\Omega $ should be at least equal to the largest geometric multiplicity of eigenvalues of $\mathbf{A}$, and we consider the universal spatiotemporal sampling sets $(\Omega, l_i)$ for convolution operators $\mathbf{A}$ with eigenvalues subject to the same largest geometric multiplicity. We will give an algebraic characterization for such sampling sets and show how this problem is linked with sparse signal processing theory and polynomial interpolation theory.