A Generalized Prony Method for Filter Recovery in Evolutionary System via Spatiotemporal Trade Off

We consider the problem of spatiotemporal sampling in an evolutionary process $x^{(n)}=A^nx$ where an unknown linear operator $A$ driving an unknown initial state $x$ is to be recovered from a combined set of coarse spatial samples $\{x|_{\Omega_0}, x^{(1)}|_{\Omega_1},\cdots, x^{(N)}|_{\Omega_N}\}$. In this paper, we will study the case of infinite dimensional spatially invariant evolutionary process, where the unknown initial signals $x $ are modeled as $\ell^2(\mathbb{Z})$ and $A$ is an unknown spatial convolution operator given by a filter $a \in \ell^1(\mathbb{Z})$ so that $Ax=a*x$. We show that $\{x|_{\Omega_m}, x^{(1)}|_{\Omega_m},\cdots, x^{(N)}|_{\Omega_m}: N \geq 2m-1, \Omega_m=m\mathbb{Z}\}$ contains enough information to recover the Fourier spectrum of a typical low pass filter $a$, if the initial signal $x$ is from a dense subset of $\ell^2(\mathbb{Z})$. The idea is based on a nonlinear, generalized Prony method similar to \cite{AK14}. We provide an algorithm for the case when both $a$ and $x$ are compactly supported around the center. Finally, we perform the accuracy analysis based on the spectral properties of the operator $A$ and the initial state $x$, and verify them by several numerical experiments.