Frames for source recovery from non-uniform dynamical samples

TL;DR

This paper studies the stability and recovery conditions of source terms in non-uniform dynamical samples within infinite-dimensional Hilbert spaces, extending prior spectral pair results.

Contribution

It provides necessary and sufficient conditions for source recovery and stability in finite and infinite iterations, generalizing previous work.

Findings

Necessary and sufficient conditions for source recovery in finitely many iterations.

Necessary condition for stability of the source term in infinite-dimensional spaces.

Conditions for source recovery in infinitely many iterations.

Abstract

Motivated by the work of Aldroubi et al., we investigate the stability of the source term of the discrete dynamical system indexing over a non-uniform discrete set arising from spectral pairs in infinite-dimensional separable Hilbert spaces. Extending results due to Aldroubi et al., firstly, we give a necessary and sufficient condition for the recovery of the source term in finitely many iterations. Afterwards, we derive a necessary condition for the stability of the source term in finitely many iterations when it belongs to the closed subspace of an infinite-dimensional separable Hilbert space. Finally, we give a necessary and sufficient condition for the recovery of the source term in infinitely many iterations.