Approximate frame representations via iterated operator systems
Volume 263 / 2022
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.
