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

We consider the problem of spatiotemporal sampling in which an initial state of an evolution process is to be recovered from a combined set of coarse spatial samples from varying time levels . This new way of sampling, which we call dynamical sampling, differs from standard sampling since at any fixed time there are not enough samples to recover the function or the state . Although dynamical sampling is an inverse problem, it differs from the typical inverse problems in which is to be recovered from for a single time . In this paper, we consider signals that are modeled by or a shift invariant space , and are evolving under the action of a spatial convolution operator , so that . 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.