|
Frames and the Feichtinger conjecture
Author(s):
Peter
G.
Casazza;
Ole
Christensen;
Alexander
M.
Lindner;
Roman
Vershynin
Journal:
Proc. Amer. Math. Soc.
133
(2005),
1025-1033.
MSC (2000):
Primary 46C05, 46L05;
Secondary 42C40
Posted:
November 19, 2004
Retrieve article in:
PDF DVI PostScript
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.
References:
-
- [BT1]
- J. Bourgain and L. Tzafriri: Invertibility of ``large'' submatricies with applications to the geometry of Banach spaces and harmonic analysis. Israel J. Math., 57 (1987) 137-223. MR 0890420 (89a:46035)
- [BT2]
- J. Bourgain, L. Tzafriri, On a problem of Kadison and Singer, J. Reine Angew. Math. 420 (1991), 1-43. MR 1124564 (92j:46104)
- [Ca]
- P.G. Casazza, The art of frame theory. Taiwanese Journal of Math., 4 (2) (2000) 129-202. MR 1757401 (2001f:42046)
- [Ch]
- O. Christensen, An introduction to frames and Riesz bases, Birkhäuser, 2002. MR 1946982 (2003k:42001)
- [CL]
- O. Christensen and A.M. Lindner, Decomposition of Riesz frames and wavelets into a finite union of linearly independent sets, Lin. Alg. Appl. 355 no. 1 (2002), 147-160. MR 1930142 (2003g:42057)
- [DS]
- K. Davidson, S. Szarek, Local operator theory, random matrices and Banach spaces, Handbook of the geometry of Banach spaces, Vol. I, 317-366, North-Holland, Amsterdam, 2001. MR 1863696 (2004f:47002a)
- [G]
- K. Gröchenig, Localized frames are finite unions of Riesz sequences. Adv. Comp. Math. 18 no. 2-4 (2003), 149-157. MR 1968117 (2004a:42044)
- [KS]
- R. Kadison and I. Singer, Extensions of pure states, American Jour. Math. 81 (1959), 547-564. MR 0123922 (23:A1243)
- [RS]
- A. Ron and Z. Shen, Weyl-Heisenberg systems and Riesz bases in
 , Duke Math. Jour. 89 (1997), 237-282. MR 1460623 (98i:42013) - [W]
- N. Weaver, The Kadison-Singer conjecture in discrepancy theory, Preprint.
- [Y]
- R. Young, An introduction to nonharmonic Fourier series - revised first edition, Academic Press, 2001. MR 1836633 (2002b:42001)
Similar Articles:
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:
10.1090/S0002-9939-04-07594-X
PII:
S 0002-9939(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
Posted:
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
Copyright of article:
Copyright
2004,
American Mathematical Society
|
Comments: webmaster@ams.org