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A decomposition theorem for frames and the Feichtinger Conjecture
Author(s):
Peter
G.
Casazza;
Gitta
Kutyniok;
Darrin
Speegle;
Janet
C.
Tremain
Journal:
Proc. Amer. Math. Soc.
136
(2008),
2043-2053.
MSC (2000):
Primary 46C05, 42C15, 46L05
Posted:
February 21, 2008
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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  -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  -independent for  -sequences.
References:
-
- 1.
- J. Anderson, Extensions, restrictions, and representations of states on
 -algebras, Trans. Amer. Math. Soc. 249 (1979), 303-329. MR 525675 (80k:46069) - 2.
- 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), 137-224. MR 890420 (89a:46035)
- 3.
- J. Bourgain and L. Tzafriri, On a problem of Kadison and Singer, J. Reine Angew. Math. 420 (1991), 1-43. MR 1124564 (92j:46104)
- 4.
- M. Bownik and D. Speegle, The Feichtinger conjecture for wavelet frames, Gabor frames and frames of translates, Canad. J. Math. 58 (2006), 1121-1143. MR 2270922 (2007h:42044)
- 5.
- P. G. Casazza, The art of frame theory, Taiwanese J. of Math. 4 (2000), 129-201. MR 1757401 (2001f:42046)
- 6.
- P. G. Casazza, O. Christensen, A. Lindner, and R. Vershynin, Frames and the Feichtinger conjecture, Proc. Amer. Math. Soc. 133 (2005), 1025-1033. MR 2117203 (2006a:46024)
- 7.
- P. G. Casazza, O. Christensen, and N. Kalton, Frames of translates, Collect. Math. 52 (2001), 35-54. MR 1833085 (2002f:42002)
- 8.
- P. G. Casazza, M. Fickus, J. C. Tremain, and E. Weber, The Kadison-Singer problem in mathematics and engineering: a detailed account, in: Operator Theory, Operator Algebras, and Applications, D. Han, P. Jorgensen, and D. R. Larson, eds., Contemp. Math. 414, Amer. Math. Soc., Providence, RI (2006), 299-356. MR 2277219 (2007j:42016)
- 9.
- P. G. Casazza and G. Kutyniok, Frames of subspaces, in: Wavelets, Frames, and Operator Theory, C. Heil, P. E. T. Jorgensen, and D. R. Larson, eds., Contemp. Math. 345, Amer. Math. Soc., Providence, RI (2004), 87-113. MR 2066823 (2005e:42090)
- 10.
- P. G. Casazza, G. Kutyniok, and S. Li, Fusion Frames and Distributed Processing, Appl. Comput. Harmon. Anal. (to appear).
- 11.
- P. G. Casazza and J. C. Tremain, The Kadison-Singer problem in mathematics and engineering, Proc. Natl. Acad. Sci. USA 103 (2006), 2032-2039. MR 2204073 (2006j:46074)
- 12.
- P. G. Casazza and R. Vershynin, Kadison-Singer meets Bourgain-Tzafriri, preprint (2005).
- 13.
- O. Christensen, An Introduction to Frames and Riesz Bases, Birkhäuser, Boston, 2003. MR 1946982 (2003k:42001)
- 14.
- O. Christensen and A. Lindner, Decomposition of Riesz frames and wavelets into a finite union of linearly independent sets, Lin. Alg. Appl. 355 (2002), 147-159. MR 1930142 (2003g:42057)
- 15.
- K. Gröchenig, Localized frames are finite unions of Riesz sequences, Adv. Comput. Math. 18 (2003), 149-157. MR 1968117 (2004a:42044)
- 16.
- R. Kadison and I. Singer, Extensions of pure states, American Jour. Math. 81 (1959), 383-400. MR 0123922 (23:A1243)
- 17.
- J. Lindenstrauss and L. Tzafriri, Classical Banach Spaces I: Sequence Spaces, Springer-Verlag, Berlin, 1977. MR 0500056 (58:17766)
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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:
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
Posted:
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
Copyright of article:
Copyright
2008,
American Mathematical Society
The copyright for this article reverts to public domain after 28 years from publication.
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