Continuous and discrete dynamical sampling

TL;DR

This paper investigates conditions under which continuous and discrete dynamical sampling generate frames in complex Hilbert spaces, exploring the transition from continuous to discrete time sampling and their equivalence.

Contribution

It provides necessary and sufficient conditions for continuous dynamical sampling to form semi-continuous frames and examines discretization and relation to discrete iterations.

Findings

Characterization of operators and sets generating semi-continuous frames.

Conditions under which discretization preserves the frame property.

Relation between continuous evolution and discrete iteration in dynamical sampling.

Abstract

In this paper we study the continuous dynamical sampling problem at infinite time in a complex Hilbert space $\mathcal{H}$. We find necessary and sufficient conditions on a bounded linear operator $A\in\mathcal{B}(\mathcal{H})$ and a set of vectors $\mathcal{G}\subset \mathcal{H}$, in order to obtain that $\{e^{tA}g\}_{g\in\mathcal{G}, t\in[0,\infty)}$ is a semi-continuous frame for $\mathcal{H}$. We study if it is possible to discretize the time variable $t$ and still have a frame for $\mathcal{H}$. We also relate the continuous iteration $e^{tA}$ on a set $\mathcal{G}$ to the discrete iteration $(A^\prime)^n$ on $\mathcal{G}^\prime$ for an adequate operator $A^\prime$ and set $\mathcal{G}^\prime\subset \mathcal{H}$.