Stable Initial-State Recovery from Dynamical Samples using Nagy-Type and Pollard-Hilding-Type Frame Perturbations

Abstract

Stimulated by Aldroubi and his collaborator's recent work on dynamical sampling, we consider a homogeneous discrete dynamical system of the form $f_n=A{f}_{n-1}=A^{n}f,f_0=f$, where A is a bounded linear operator on a separable Hilbert space H, which is known as the evolution operator, and $f_0\in \mathcal{H}$ is the unknown initial state. The associated dynamical samples are given by the collection $\{⟨A^{n}f,g⟩:g\in \mathcal{G},0\le n<n<\mathcal{L}(g)\}$, where $\mathcal{G}\subset H$, is a finite or countable sampling set and $L$ is a function $L:G\to \mathbb{N}\bigcup \{\infty \}$. We analyze the stability of perturbed dynamical sampling systems in the sense of Nagy and Pollard-Hilding. More precisely, we establish sufficient conditions for the stable recovery of an initial state from perturbed dynamical samples obtained by changing the sampling vector, the evolution operator, ro simultaneously both, within the framework of Nagy-type and Pollard-Hilding-type perturbation of frames.