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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

A decomposition theorem for frames and the Feichtinger Conjecture


Authors: Peter G. Casazza, Gitta Kutyniok, Darrin Speegle and Janet C. Tremain
Journal: Proc. Amer. Math. Soc. 136 (2008), 2043-2053
MSC (2000): Primary 46C05, 42C15, 46L05
Published electronically: February 21, 2008
MathSciNet review: 2383510
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Abstract: In this paper we study the Feichtinger Conjecture in frame theory, which was recently shown to be equivalent to the 1959 Kadison-Singer Problem in $ C^{*}$-Algebras. We will show that every bounded Bessel sequence can be decomposed into two subsets each of which is an arbitrarily small perturbation of a sequence with a finite orthogonal decomposition. This construction is then used to answer two open problems concerning the Feichtinger Conjecture: 1. The Feichtinger Conjecture is equivalent to the conjecture that every unit norm Bessel sequence is a finite union of frame sequences. 2. Every unit norm Bessel sequence is a finite union of sets each of which is $ \omega$-independent for $ \ell_2$-sequences.


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Additional Information

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

Gitta Kutyniok
Affiliation: Program in Applied and Computational Mathematics, Princeton University, Princeton, New Jersey 08544
Email: kutyniok@math.princeton.edu

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

Janet C. Tremain
Affiliation: Department of Mathematics, University of Missouri, Columbia, Missouri 65211
Email: janet@math.missouri.edu

DOI: http://dx.doi.org/10.1090/S0002-9939-08-09264-2
PII: S 0002-9939(08)09264-2
Keywords: Bessel sequence, decomposition, frame, Feichtinger Conjecture, frame sequence, Kadison-Singer Conjecture, $\omega$-independence, Riesz basic sequence
Received by editor(s): January 25, 2007
Published electronically: February 21, 2008
Additional Notes: The first author was supported by NSF Grant DMS 0405376.
The second author was supported by Deutsche Forschungsgemeinschaft (DFG) Research Fellowship KU 1446/5-1, by Preis der Justus-Liebig-Universität Gießen 2006, and by Deutsche Forschungsgemeinschaft (DFG) Heisenberg-Fellowship KU 1446/8-1.
The third author was supported by NSF Grant DMS 0354957.
Communicated by: N. Tomczak-Jaegermann
Article copyright: © Copyright 2008 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.