Operator representations of sequences and dynamical sampling

TL;DR

This paper advances the theory of dynamical sampling by exploring operator-based representations of sequences in Hilbert spaces and providing a new proof for a class of frames representable via bounded operators.

Contribution

It generalizes operator representations of frames and offers a new proof for a class of frames that can be represented through bounded operators.

Findings

Sequences in Hilbert spaces can be represented via iterated bounded operators.

A new proof demonstrates a class of frames are representable by bounded operators.

The approach may lead to broader classes of frames with operator representations.

Abstract

This paper is a contribution to the theory of dynamical sampling. Our purpose is twofold. We first consider representations of sequences in a Hilbert space in terms of iterated actions of a bounded linear operator. This generalizes recent results about operator representations of frames, and is motivated by the fact that only very special frames have such a representation. As our second contribution we give a new proof of a construction of a special class of frames that are proved by Aldroubi et al. to be representable via a bounded operator. Our proof is based on a single result by Shapiro \& Shields and standard frame theory, and our hope is that it eventually can help to provide more general classes of frames with such a representation.