Dynamical Sampling on Finite Index Sets

We consider bounded operators $A$ acting iteratively on a finite set of vectors $\{f_i : i\in I\}$ in a Hilbert space $\mathcal H$ and address the problem of providing necessary and sufficient conditions for the collection of iterates $\{A^nf_i : i\in I,\,n=0,1,2,\ldots\}$ to form a frame for the space $\mathcal H$. For normal operators $A$ we completely solve the problem by proving a characterization theorem. Our proof incorporates techniques from different areas of mathematics, such as operator theory, spectral theory, harmonic analysis, and complex analysis in the unit disk. In the second part of the paper we drop the strong condition on $A$ to be normal. Despite this quite general setting, we are able to prove a characterization which allows to infer many strong necessary conditions on the operator $A$. For example, $A$ needs to be similar to a contraction of a very special kind. We also prove a characterization theorem for the finite-dimensional case. --- These results provide a theoretical solution to the so-called Dynamical Sampling problem where a signal $f$ that is evolving in time through iterates of an operator $A$ is spatially sub-sampled at various times and one seeks to reconstruct the signal $f$ from these spatial-temporal samples.