# Frame representations via suborbits of bounded operators

Ole Christensen, Marzieh Hasannasab

Research output: Chapter in Book/Report/Conference proceedingArticle in proceedingsResearchpeer-review

## Abstract

The standard setup of dynamical sampling concerns frame properties of sequences of the form $\left\{ {{T^n}\varphi } \right\}_{n = 0}^\infty$, where T is a bounded operator on a Hilbert space $\mathcal{H}$ and $\varphi \in \mathcal{H}$. In this paper we consider two generalizations of this basic idea. We first show that the class of frames that can be represented using iterations of a bounded operator increases drastically if we allow representations using just a subfamily $\left\{ {{T^{\alpha (k)}}\varphi } \right\}_{n = 0}^\infty$ of $\left\{ {{T^n}\varphi } \right\}_{n = 0}^\infty$; indeed, any linear independent frame has such a representation for a certain bounded operator T. Furthermore, we prove a number of results relating the properties of the frame and the distribution of the powers $\{ \alpha (k)\} _{k = 1}^\infty$ in $\mathbb{N}$. Finally we show that also the condition of linear independency can be removed by considering approximate frame representations with an arbitrary small prescribed tolerance, in a sense to be made precise.
Original language English Proceedings of 2019 13th International conference on Sampling Theory and Applications IEEE 2019 1-4 9781728137414 https://doi.org/10.1109/sampta45681.2019.9030900 Published - 2019 2019 13th International conference on Sampling Theory and Applications - University of Bordeaux, FranceDuration: 8 Jul 2019 → 12 Jul 2019

### Conference

Conference 2019 13th International conference on Sampling Theory and Applications France University of Bordeaux 08/07/2019 → 12/07/2019