A survey on frame representations via dynamical sampling (2201.00038v1)

Abstract: 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.