Dynamical sampling and systems from iterative actions of operators

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.