A survey on frame representations via dynamical sampling
| Autor: | Christensen, Ole, Hasannasab, Marzieh |
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| Rok vydání: | 2021 |
| Předmět: | |
| Druh dokumentu: | Working Paper |
| Popis: | 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. |
| Databáze: | arXiv |
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