  Dynamical sampling deals with representations of a frame $\{ f_k
\}_{k=1}^\infty$ as an orbit $\{ T^n \varphi \}_{n=0}^\infty$ of a linear and
possibly bounded operator $T$ acting on the underlying Hilbert space. It is
known that the desire of boundedness of the operator $T$ puts severe
restrictions on the frame $\{ f_k \}_{k=1}^\infty$. The purpose of the paper is
to present an overview of the results in the literature and also discuss
various alternative ways of representing a frame; in particular the class of
considered frames can be enlarged drastically by allowing representations using
only a subset $\{ T^{\alpha(k)} \varphi\}^\infty_{k=1}$ of the operator orbit
$\{ T^n \varphi \}_{n=0}^\infty$. In general it is difficult to specify
appropriate values for the scalars $\alpha(k)$ and the vector $\varphi;$
however, by accepting an arbitrarily small and controllable deviation between
the given frame $\{ f_k \}_{k=1}^\infty$ and $\{ T^{\alpha(k)}
\varphi\}_{k=1}^\infty$ we will be able to do so.
