Characterization of frames for source recovery from dynamical samples

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.