Frame representations via suborbits of bounded operators

The standard setup of dynamical sampling concerns frame properties of sequences of the form {T
ⁿ
φ}
^∞
_n=0
, where T is a bounded operator on a Hilbert space H and φ ϵ 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 {T
^α(k)
φ}
^∞
_n=0
of {T
ⁿ
φ}
^∞n=0
; 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 {α(k)}
^∞
_k=1
in 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.