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

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 $\{A^{n}g{\}}_{g\in \mathcal{G},0\le n\le L}$ arising in the dynamical sampling problem.