基于动态抽样的框架表示方法综述
A survey on frame representations via dynamical sampling
摘要：
动态采样处理表示的框架$\{ f_k \}_{k=1}^\infty$作为一个轨道$\{ T^n φ\}_{n=0}^\infty$的线性和可能有界算子$T$上的基础希尔伯特空间。已知算子$T$的有界性要求对框架$\{ f_k \}_{k=1}^\infty$施加了严格的限制。本文的目的是概述文献中的结果，并讨论表示框架的各种替代方法;特别是，通过允许仅使用算子轨道的子集$\{ T^{α(k)} φ\}^\infty_{k=1}$进行表示，可以大幅扩大所考虑的框架类别$\{ T^n φ\}_{n=0}^\infty$。一般来说，很难为标量$α(k)$和矢量$φ;$指定适当的值，然而，通过接受给定帧$\{ f_k \}_{k=1}^\infty$和$\{ T^{α(k)} φ\}_{k=1}^\infty$之间的任意小且可控的偏差，我们将能够这样做。
英文摘要：
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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