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Spanning and independence properties of frame partitions


Authors: Bernhard G. Bodmann, Peter G. Casazza, Vern I. Paulsen and Darrin Speegle
Journal: Proc. Amer. Math. Soc. 140 (2012), 2193-2207
MSC (2010): Primary 15A03, 42C15
DOI: https://doi.org/10.1090/S0002-9939-2011-11072-4
Published electronically: October 24, 2011
MathSciNet review: 2898683
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Abstract: We answer a number of open problems in frame theory concerning the decomposition of frames into linearly independent and/or spanning sets. We prove that Parseval frames with norms bounded away from $ 1$ can be decomposed into a number of sets whose complements are spanning, where the number of these sets only depends on the norm bound. Further, we prove a stronger result for Parseval frames whose norms are uniformly small, which shows that in addition to the spanning property, the sets can be chosen to be independent and the complement of each set can contain a number of disjoint, spanning sets.


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Additional Information

Bernhard G. Bodmann
Affiliation: Department of Mathematics, 651 Philip G. Hoffman Hall, University of Houston, Houston, Texas 77204-3008
Email: bgb@math.uh.edu

Peter G. Casazza
Affiliation: Department of Mathematics, University of Missouri, Columbia, Missouri 65211
Email: casazzap@missouri.edu

Vern I. Paulsen
Affiliation: Department of Mathematics, 651 Philip G. Hoffman Hall, University of Houston, Houston, Texas 77204-3008
Email: vern@math.uh.edu

Darrin Speegle
Affiliation: Department of Mathematics and Computer Science, Saint Louis University, St. Louis, Missouri 63103
Email: speegled@slu.edu

DOI: https://doi.org/10.1090/S0002-9939-2011-11072-4
Received by editor(s): October 29, 2010
Received by editor(s) in revised form: February 14, 2011
Published electronically: October 24, 2011
Additional Notes: The first author was supported by NSF grant DMS-0807399
The second author was supported by NSF 1008183: DTRA/NSF 1042701
The third author was supported by NSF DMS-0600191
The fourth author was supported by NSF DMS-0354957
Communicated by: Thomas Schlumprecht
Article copyright: © Copyright 2011 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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