Exact Reconstruction of Signals in Evolutionary Systems Via Spatiotemporal Trade-off

Abstract

We consider the problem of spatiotemporal sampling in which an initial state $f$ of an evolution process $f_t=A_tf$ is to be recovered from a combined set of coarse spatial samples from varying time levels $\{t_1,\dots ,t_N\}$. This new way of sampling, which we call dynamical sampling, differs from standard sampling since at any fixed time $t_i$ there are not enough samples to recover the function $f$ or the state $f_{t_i}$. Although dynamical sampling is an inverse problem, it differs from the typical inverse problems in which $f$ is to be recovered from $A_Tf$ for a single time $T$. In this paper, we consider signals that are modeled by $\ell ^2({\mathbb Z})$ or a shift invariant space $V\subset L^2({\mathbb R})$, and are evolving under the action of a spatial convolution operator $A$, so that $f_{n}=A^nf$. We provide sufficient conditions for the spatiotemporal sampling problem to be solvable. In special cases, we provide error analysis based on the spectral properties of the operator $A$.