Approximate frame representations via iterated operator systems

Ole Christensen, Marzieh Hasannasab

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Abstract

It is known that it is a very restrictive condition for a frame $\{f_k\}_{k=1}^\infty$ to have a representation $ \{T^n \varphi\}_{n=0}^\infty$ as the orbit of a bounded operator $T$ under a single generator $\varphi\in\mathcal{H}.$ In this paper we prove that, on the other hand, any frame can be approximated arbitrarily well by a suborbit $\{T^{\alpha(k)} \varphi\}_{k=1}^\infty$ of a bounded operator $T$. An important new aspect is that for certain important classes of frames, e.g., frames consisting of finitely supported vectors in $\ell^2(\mathbb{N}),$ we can be completely explicit about possible choices of the operator $T$ and the powers $\alpha(k),k\in \mathbb{N}.$ A similar approach carried out in $L^2(\mathbb{R})$ leads to an approximation of a frame using suborbits of two bounded operators. The results are illustrated with an application to Gabor frames generated by a compactly supported function. The paper is concluded with an appendix which collects general results about frame representations using multiple orbits of bounded operators.
Original languageEnglish
JournalStudia Mathematica
Number of pages19
ISSN0039-3223
Publication statusAccepted/In press - 2020

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