A survey on frame representations via dynamical sampling

Dynamical sampling deals with representations of a frame $\{ f_k \}_{k=1}^\infty$ as an orbit $\{ T^n φ\}_{n=0}^\infty$ 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 $\{ T^{α(k)} φ\}^\infty_{k=1}$ of the operator orbit $\{ T^n φ\}_{n=0}^\infty$. In general it is difficult to specify appropriate values for the scalars $α(k)$ and the vector $φ;$ however, by accepting an arbitrarily small and controllable deviation between the given frame $\{ f_k \}_{k=1}^\infty$ and $\{ T^{α(k)} φ\}_{k=1}^\infty$ we will be able to do so.

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