Frame representations via suborbits of bounded operators

The standard setup of dynamical sampling concerns frame properties of sequences of the form $\left\{ {{T^n}\varphi } \right\}_{n = 0}^\infty $, where T is a bounded operator on a Hilbert space $\mathcal{H}$ and $\varphi \in \mathcal{H}$. In this paper we consider two generalizations of this basic idea. We first show that the class of frames that can be represented using iterations of a bounded operator increases drastically if we allow representations using just a subfamily $\left\{ {{T^{\alpha (k)}}\varphi } \right\}_{n = 0}^\infty $ of $\left\{ {{T^n}\varphi } \right\}_{n = 0}^\infty $; indeed, any linear independent frame has such a representation for a certain bounded operator T. Furthermore, we prove a number of results relating the properties of the frame and the distribution of the powers $\{ \alpha (k)\} _{k = 1}^\infty $ in $\mathbb{N}$. Finally we show that also the condition of linear independency can be removed by considering approximate frame representations with an arbitrary small prescribed tolerance, in a sense to be made precise.