Frames and the Feichtinger conjecture

Authors:
Peter G. Casazza, Ole Christensen, Alexander M. Lindner and Roman Vershynin

Journal:
Proc. Amer. Math. Soc. **133** (2005), 1025-1033

MSC (2000):
Primary 46C05, 46L05; Secondary 42C40

DOI:
https://doi.org/10.1090/S0002-9939-04-07594-X

Published electronically:
November 19, 2004

MathSciNet review:
2117203

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Abstract | References | Similar Articles | Additional Information

Abstract: We show that the conjectured generalization of the Bourgain-Tzafriri *restricted-invertibility theorem* is equivalent to the conjecture of Feichtinger, stating that every bounded frame can be written as a finite union of Riesz basic sequences. We prove that any bounded frame can at least be written as a finite union of linearly independent sequences. We further show that the two conjectures are implied by the *paving conjecture*. Finally, we show that Weyl-Heisenberg frames over rational lattices are finite unions of Riesz basic sequences.

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Additional Information

**Peter G. Casazza**

Affiliation:
Department of Mathematics, University of Missouri-Columbia, Columbia, Missouri 65211

Email:
pete@math.missouri.edu

**Ole Christensen**

Affiliation:
Department of Mathematics, Technical University of Denmark, Building 303, 2800 Lyngby, Denmark

Email:
Ole.Christensen@mat.dtu.dk

**Alexander M. Lindner**

Affiliation:
Center of mathematical Sciences, Munich University of Technology, Boltzmannstr. 3, D-85747 Garching, Germany

Email:
lindner@mathematik.tu-muenchen.de

**Roman Vershynin**

Affiliation:
Department of Mathematics, University of California at Davis, One Shields Avenue, Davis, California 95016

Email:
vershynin@math.ucdavis.edu

DOI:
https://doi.org/10.1090/S0002-9939-04-07594-X

Keywords:
Kadison-Singer problem,
paving conjecture,
Feichtinger conjecture,
frames

Received by editor(s):
February 18, 2003

Received by editor(s) in revised form:
July 3, 2003

Published electronically:
November 19, 2004

Additional Notes:
The first author was supported by NSF DMS 0102686

The last author thanks PIMS for support

Communicated by:
David R. Larson

Article copyright:
© Copyright 2004
American Mathematical Society