Exact Reconstruction of Spatially Undersampled Signals in Evolutionary Systems

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 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 $V\subset L^2(\mathbb R)$.