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Approximate frame representations via iterated operator systems

Volume 263 / 2022

Ole Christensen, Marzieh Hasannasab Studia Mathematica 263 (2022), 1-18 MSC: Primary 42C15. DOI: 10.4064/sm190228-5-8 Published online: 22 November 2021

Abstract

It is known that it is a very restrictive condition for a frame $\{f_{k}\}^{\infty }_{k=1}$ to have a representation $ \{T^n \varphi \}_{n=0}^\infty $ as the orbit of a bounded operator $T$ under a single generator $\varphi \in \mathcal H.$ 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.

Authors

  • Ole ChristensenTechnical University of Denmark
    DTU Compute
    Building 303
    2800 Lyngby, Denmark
    e-mail
  • Marzieh HasannasabInstitut für Mathematik
    Technical University of Berlin
    Straße des 17. Juni 136
    10623 Berlin, Germany
    e-mail

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