  This paper is a contribution to the theory of dynamical sampling. Our purpose
is twofold. We first consider representations of sequences in a Hilbert space
in terms of iterated actions of a bounded linear operator. This generalizes
recent results about operator representations of frames, and is motivated by
the fact that only very special frames have such a representation. As our
second contribution we give a new proof of a construction of a special class of
frames that are proved by Aldroubi et al. to be representable via a bounded
operator. Our proof is based on a single result by Shapiro \& Shields and
standard frame theory, and our hope is that it eventually can help to provide
more general classes of frames with such a representation.
