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 $f_t=A_{t}f$ 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 $A_{T}f$ for a single time $T$. In this paper, we consider signals that are modeled by ${\ell }^2(\mathbb{Z})$ or a shift invariant space $V\subset L^2(\mathbb{R})$.

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