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Operator representations of frames: boundedness, duality, and stability
The purpose of the paper is to analyze frames
having the form for some linear operator T:
\mbox{span} \{f_k\}_{k\in \mathbf Z} \to \mbox{span}\{f_k\}_{k\in \mathbf Z}.
A key result characterizes boundedness of the operator in terms of
shift-invariance of a certain sequence space. One of the consequences is a
characterization of the case where the representation can be achieved for an operator that has an
extension to a bounded bijective operator
In this case we also characterize all the dual frames that are representable in
terms of iterations of an operator in particular we prove that the only
possible operator is Finally, we consider stability
of the representation 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
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