Operator representations of frames

Ole Christensen, Marzieh Hasannasab

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

Abstract

The purpose of this paper is to consider representations of frames {fk}k∈I in a Hilbert space ℋ of the form {fk}k∈I = {Tkf0}k∈I for a linear operator T; here the index set I is either ℤ or ℒ0. While a representation of this form is available under weak conditions on the frame, the analysis of the properties of the operator T requires more work. For example it is a delicate issue to obtain a representation with a bounded operator, and the availability of such a representation not only depends on the frame considered as a set, but also on the chosen indexing. Using results from operator theory we show that by embedding the Hilbert space ℋ into a larger Hilbert space, we can always represent a frame via iterations of a bounded operator, composed with the orthogonal projection onto ℋ. The paper closes with a discussion of an open problem concerning representations of Gabor frames via iterations of a bounded operator.
Original languageEnglish
Title of host publicationProceedings of 2017 international conference on sampling theory and applications
PublisherIEEE
Publication date2017
Pages207-11
ISBN (Print)978-1-5386-1565-2
DOIs
Publication statusPublished - 2017
Event2017 International Conference on Sampling Theory and Applications - Tallinn, Estonia
Duration: 3 Jul 20177 Jul 2017

Conference

Conference2017 International Conference on Sampling Theory and Applications
CountryEstonia
CityTallinn
Period03/07/201707/07/2017

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