Approximate frame representations via iterated operator systems

TL;DR

This paper demonstrates that any frame in a Hilbert space can be approximated arbitrarily well by suborbits of bounded operators, providing explicit constructions for certain classes like finitely supported vectors and Gabor frames.

Contribution

It introduces a method to approximate arbitrary frames using suborbits of bounded operators, expanding the understanding of frame representations beyond exact conditions.

Findings

Any frame can be approximated by suborbits of a bounded operator.

Explicit constructions are provided for finitely supported vectors in ().

Application to Gabor frames with compactly supported functions.

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.