Approximate frame representations via iterated operator systems
Volume 263 / 2022
Abstract
It is known that it is a very restrictive condition for a frame 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.