Approximate frame representations via iterated operator systems

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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, anyframe can be approximated arbitrarily well by a suborbit $\{T^{\alpha(k)}\varphi\}_{k=1}^\infty$ of a bounded operator $T$. An important new aspect isthat for certain important classes of frames, e.g., frames consisting offinitely supported vectors in $\ell^2(\mathbb{N}),$ we can be completelyexplicit 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 twobounded operators. The results are illustrated with an application to Gaborframes generated by a compactly supported function. The paper is concluded withan appendix which collects general results about frame representations usingmultiple orbits of bounded operators.
Original languageEnglish
JournalStudia Mathematica
Number of pages19
Publication statusAccepted/In press - 2023


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