Dynamical sampling, frames, and inverse problems
Dynamical sampling is a term describing an emerging set of problems related to recovering signals and evolution operators from space-time samples. For example, consider the abstract IVP in a separable Hilbert space H:
where t∈[0,∞), u:R+→H,
˙u:R+→H is the time derivative of u, and u0 is an initial condition. When, F=0, A is a known (or unknown) operator, and the goal is to recover u0 from the samples {u(ti,xj)} on a sampling set {(ti,xj)}, we get the so called {\em space-time sampling} problems. If the goal is to identify the operator A, or some of its characteristics, we get the {\em system identification} problems. If instead we wish to recover F, we get the {\em source term} problems. In this talk, I will present several results on dynamical sampling, their connection to frames and inverse problems.