Dynamical sampling and systems from iterative actions of operators

TL;DR

This paper reviews recent advances and presents new results on frames generated by iterative actions of operators, with applications to dynamical sampling for reconstructing functions from space-time samples.

Contribution

It introduces new theoretical results on frames and Bessel systems formed by operator iterations, advancing the understanding of dynamical sampling in Hilbert spaces.

Findings

New conditions for frames generated by operator iterations

Extensions of dynamical sampling theory

Results on stability and reconstruction from space-time samples

Abstract

We review some of the recent developments and prove new results concerning frames and Bessel systems generated by iterations of the form $\{A^ng: g\in G,\, n=0,1,2,\dots \}$, where $A$ is a bounded linear operators on a separable complex Hilbert space $ \mathcal{H} $ and $G$ is a countable set of vectors in $ \mathcal{H} $. The system of iterations mentioned above was motivated from the so called dynamical sampling problem. In dynamical sampling, an unknown function $f$ and its future states $A^nf$ are coarsely sampled at each time level $n$, $0\leq n< L$, where $A$ is an evolution operator that drives the system. The goal is to recover $f$ from these space-time samples.