Periodic source detection in discrete dynamical systems via space–time sampling

In this paper, we examine a discrete dynamical system defined by $x(n+1)=Ax\left(n\right)+w\left(n\right)$, 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 $G={\left\{g_j\right\}}_{j\in J}$ of $H$ that enable stable recovery of the unknown source term $w$ from space–time samples ${\left\{〈x\left(n\right),g_j〉\right\}}_{n\ge 0,j\in J}$. We provide necessary and sufficient conditions on $G={\left\{g_j\right\}}_{j\in J}$ to ensure stable recovery of any $w\in W$. Additionally, we explicitly construct an operator $R$, dependent on $G$, such that $R{\left\{〈x\left(n\right),g_j〉\right\}}_{n,j}=w$.