Year: 2021
Type: preprint
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.
Source: arXiv (Cornell University)
Authors: Christensen, Ole, Hasannasab, Marzieh
Language: English
Subfield: Applied Mathematics
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W4225879167