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

Published electronically:
November 19, 2004

MathSciNet review:
2117203

Full-text PDF Free Access

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.

**[BT1]**J. Bourgain and L. Tzafriri,*Invertibility of “large” submatrices with applications to the geometry of Banach spaces and harmonic analysis*, Israel J. Math.**57**(1987), no. 2, 137–224. MR**890420**, 10.1007/BF02772174**[BT2]**J. Bourgain and L. Tzafriri,*On a problem of Kadison and Singer*, J. Reine Angew. Math.**420**(1991), 1–43. MR**1124564****[Ca]**Peter G. Casazza,*The art of frame theory*, Taiwanese J. Math.**4**(2000), no. 2, 129–201. MR**1757401****[Ch]**Ole Christensen,*An introduction to frames and Riesz bases*, Applied and Numerical Harmonic Analysis, Birkhäuser Boston, Inc., Boston, MA, 2003. MR**1946982****[CL]**Ole Christensen and Alexander M. Lindner,*Decomposition of Riesz frames and wavelets into a finite union of linearly independent sets*, Linear Algebra Appl.**355**(2002), 147–159. MR**1930142**, 10.1016/S0024-3795(02)00347-6**[DS]**Kenneth R. Davidson and Stanislaw J. Szarek,*Local operator theory, random matrices and Banach spaces*, Handbook of the geometry of Banach spaces, Vol. I, North-Holland, Amsterdam, 2001, pp. 317–366. MR**1863696**, 10.1016/S1874-5849(01)80010-3**[G]**Karlheinz Gröchenig,*Localized frames are finite unions of Riesz sequences*, Adv. Comput. Math.**18**(2003), no. 2-4, 149–157. Frames. MR**1968117**, 10.1023/A:1021368609918**[KS]**Richard V. Kadison and I. M. Singer,*Extensions of pure states*, Amer. J. Math.**81**(1959), 383–400. MR**0123922****[RS]**Amos Ron and Zuowei Shen,*Weyl-Heisenberg frames and Riesz bases in 𝐿₂(𝐑^{𝐝})*, Duke Math. J.**89**(1997), no. 2, 237–282. MR**1460623**, 10.1215/S0012-7094-97-08913-4**[W]**N. Weaver,*The Kadison-Singer conjecture in discrepancy theory*, Preprint.**[Y]**Robert M. Young,*An introduction to nonharmonic Fourier series*, 1st ed., Academic Press, Inc., San Diego, CA, 2001. MR**1836633**

Retrieve articles in *Proceedings of the American Mathematical Society*
with MSC (2000):
46C05,
46L05,
42C40

Retrieve articles in all journals with MSC (2000): 46C05, 46L05, 42C40

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:
http://dx.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