Characterization of frames for source recovery from dynamical samples

TL;DR

This paper investigates the conditions under which a constant source term in a discrete dynamical system can be stably recovered from time-space measurements, with implications for environmental monitoring.

Contribution

It provides necessary and sufficient conditions for source recovery in dynamical systems using Bessel system measurements, independent of initial states.

Findings

Established conditions for stable source recovery

Proved independence from initial state

Applicable to environmental monitoring scenarios

Abstract

In this paper, we address the problem of recovering constant source terms in a discrete dynamical system represented by $x_{n+1} = Ax_n + w$, where $x_n$ is the $n$-th state in a Hilbert space $\mathcal{H}$, $A$ is a bounded linear operator in $\mathcal{B}(\mathcal{H})$, and $w$ is a source term within a closed subspace $W$ of $\HH$. Our focus is on the stable recovery of $w$ using time-space sample measurements formed by inner products with vectors from a Bessel system $\mathcal{G} \subset \mathcal{H}$. We establish the necessary and sufficient conditions for the recovery of $w$ from these measurements, independent of the unknown initial state $x_0$ and for any $w \in W$. This research is particularly relevant to applications such as environmental monitoring, where precise source identification is critical.