Filter recovery in infinite spatially invariant evolutionary system via spatiotemporal trade off

We consider the problem of spatiotemporal sampling in an evolutionary process xⁿ = Aⁿx where an unknown operator A driving an unknown initial state x is to be recovered from a combined set of coarse spatial samples {χ|_Ωο, x⁽¹⁾|_Ωι,· · ·, x^(N)|_ΩN}. In this paper, we will study the case of infinite dimensional spatially invariant evolutionary process, where the unknown initial signals x are modeled as ℓ²(Z) and A is an unknown spatial convolution operator given by a filter α ε ℓ¹ (Z) so that Ax = a · x. We show that {x|Ω_m, x⁽¹⁾|Ω_m, ···, x^(N)|Ω_m:N≥2m -, Ω_m = mZ} contains enough information to recover the Fourier spectrum of a typical low pass filter a, if x is from a dense subset of ℓ² (Z). The idea is based on a nonlinear, generalized Prony method similar to [2]. We provide an algorithm for the case when a and x are both compactly supported. Finally, We perform the accuracy analysis based on the spectral properties of the operator A and the initial state x and verify them in several numerical experiments.