Periodic Source Detection in Discrete Dynamical Systems via space-time sampling

TL;DR

This paper investigates conditions under which a periodic source in a discrete dynamical system can be stably recovered from space-time samples, providing necessary and sufficient criteria and an explicit reconstruction operator.

Contribution

It introduces necessary and sufficient conditions on spatial sampling systems for stable source recovery and constructs an explicit operator for reconstruction.

Findings

Established criteria for stable source recovery from samples.

Constructed an explicit reconstruction operator R.

Provided conditions applicable to general Hilbert space settings.

Abstract

In this paper, we examine a discrete dynamical system defined by x(n+1) = Ax(n)+ w(n), where x takes values in a Hilbert space H and w is a periodic source with values in a fixed closed subspace W of H. Our goal is to identify conditions on some spatial sampling system G = {gj: j in J} of H that enable stable recovery of the unknown source term w from space-time samples {<x(n),g_j>: n >=0,j in J}. We provide necessary and sufficient conditions on G = {g_j }_{j in J} to ensure stable recovery of any w in W . Additionally, we explicitly construct an operator R, dependent on G, such that R{<x(n),g_j>}_n,j} = w.