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 $\mathcal{ G}^\prime\subset \mathcal{ H}$.
Categories: math.FA