Local-to-global frames and applications to dynamical sampling problem

TL;DR

This paper investigates conditions under which local reconstruction properties of vector systems in Hilbert spaces extend to global stable recovery, with applications to dynamical sampling systems.

Contribution

It establishes new local-to-global results for vector systems, especially those arising in dynamical sampling, enhancing understanding of stable recovery in Hilbert spaces.

Findings

Derived conditions for local-to-global stability extension.

Applied results to systems generated by powers of an operator.

Improved understanding of dynamical sampling system properties.

Abstract

In this paper we consider systems of vectors in a Hilbert space $\mathcal{H}$ of the form $\{g_{jk}: j \in J, \, k\in K\}\subset \mathcal{H}$ where $J$ and $K$ are countable sets of indices. We find conditions under which the local reconstruction properties of such a system extend to global stable recovery properties on the whole space. As a particular case, we obtain new local-to-global results for systems of type $\{A^ng\}_{g\in\mathcal{G},0\leq n\leq L }$ arising in the dynamical sampling problem.