Operator Representations of Frames: Boundedness, Duality, and Stability

Abstract

The purpose of the paper is to analyze frames $\{f_k\}_{k\in \mathbb Z}$ having the form $\{T^kf_0\}_{k\in \mathbb Z}$ for some linear operator $T: \text{ span } \{f_k\}_{k\in \mathbb Z}\rightarrow \text{ span }\{f_k\}_{k\in \mathbb Z}$. A key result characterizes boundedness of the operator T in terms of shift-invariance of a certain sequence space. One of the consequences is a characterization of the case where the representation $\{f_k\}_{k\in \mathbb Z}=\{T^kf_0\}_{k\in \mathbb Z}$ can be achieved for an operator T that has an extension to a bounded bijective operator $\widetilde{T}: {\mathcal H}\rightarrow {\mathcal H}.$ In this case we also characterize all the dual frames that are representable in terms of iterations of an operator V;  in particular we prove that the only possible operator is $V=(\widetilde{T}^*)^{-1}.$ Finally, we consider stability of the representation $\{T^kf_0\}_{k\in \mathbb Z};$ rather surprisingly, it turns out that the possibility to represent a frame on this form is sensitive towards some of the classical perturbation conditions in frame theory. Various ways of avoiding this problem will be discussed. Throughout the paper the results will be connected with the operators and function systems appearing in applied harmonic analysis, as well as with general group representations.