Frame representations via suborbits of bounded operators

Ole Christensen, Marzieh Hasannasab

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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 languageEnglish
Title of host publicationProceedings of 2019 13th International conference on Sampling Theory and Applications
PublisherIEEE
Publication date2019
Pages1-4
ISBN (Print)9781728137414
DOIs
Publication statusPublished - 2019
Event2019 13th International conference on Sampling Theory and Applications - University of Bordeaux, France
Duration: 8 Jul 201912 Jul 2019

Conference

Conference2019 13th International conference on Sampling Theory and Applications
CountryFrance
CityUniversity of Bordeaux
Period08/07/201912/07/2019

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