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

We consider the problem of spatiotemporal sampling in an evolutionary process x
ⁿ
= A
ⁿ
x 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 ℓ
²
(Z) and A is an unknown spatial convolution operator given by a filter α ε ℓ
¹
(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 ℓ
²
(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.