A decomposition theorem for frames and the Feichtinger Conjecture
HTML articles powered by AMS MathViewer
- by Peter G. Casazza, Gitta Kutyniok, Darrin Speegle and Janet C. Tremain PDF
- Proc. Amer. Math. Soc. 136 (2008), 2043-2053 Request permission
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.References
- Joel Anderson, Extensions, restrictions, and representations of states on $C^{\ast }$-algebras, Trans. Amer. Math. Soc. 249 (1979), no. 2, 303–329. MR 525675, DOI 10.1090/S0002-9947-1979-0525675-1
- 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, DOI 10.1007/BF02772174
- J. Bourgain and L. Tzafriri, On a problem of Kadison and Singer, J. Reine Angew. Math. 420 (1991), 1–43. MR 1124564, DOI 10.1515/crll.1991.420.1
- Marcin Bownik and Darrin Speegle, The Feichtinger conjecture for wavelet frames, Gabor frames and frames of translates, Canad. J. Math. 58 (2006), no. 6, 1121–1143. MR 2270922, DOI 10.4153/CJM-2006-041-3
- Peter G. Casazza, The art of frame theory, Taiwanese J. Math. 4 (2000), no. 2, 129–201. MR 1757401, DOI 10.11650/twjm/1500407227
- Peter G. Casazza, Ole Christensen, Alexander M. Lindner, and Roman Vershynin, Frames and the Feichtinger conjecture, Proc. Amer. Math. Soc. 133 (2005), no. 4, 1025–1033. MR 2117203, DOI 10.1090/S0002-9939-04-07594-X
- Peter G. Casazza, Ole Christensen, and Nigel J. Kalton, Frames of translates, Collect. Math. 52 (2001), no. 1, 35–54. MR 1833085
- Peter G. Casazza, Matthew Fickus, Janet C. Tremain, and Eric Weber, The Kadison-Singer problem in mathematics and engineering: a detailed account, Operator theory, operator algebras, and applications, Contemp. Math., vol. 414, Amer. Math. Soc., Providence, RI, 2006, pp. 299–355. MR 2277219, DOI 10.1090/conm/414/07820
- Peter G. Casazza and Gitta Kutyniok, Frames of subspaces, Wavelets, frames and operator theory, Contemp. Math., vol. 345, Amer. Math. Soc., Providence, RI, 2004, pp. 87–113. MR 2066823, DOI 10.1090/conm/345/06242
- P. G. Casazza, G. Kutyniok, and S. Li, Fusion Frames and Distributed Processing, Appl. Comput. Harmon. Anal. (to appear).
- Peter G. Casazza and Janet Crandell Tremain, The Kadison-Singer problem in mathematics and engineering, Proc. Natl. Acad. Sci. USA 103 (2006), no. 7, 2032–2039. MR 2204073, DOI 10.1073/pnas.0507888103
- P. G. Casazza and R. Vershynin, Kadison-Singer meets Bourgain-Tzafriri, preprint (2005).
- Ole Christensen, An introduction to frames and Riesz bases, Applied and Numerical Harmonic Analysis, Birkhäuser Boston, Inc., Boston, MA, 2003. MR 1946982, DOI 10.1007/978-0-8176-8224-8
- 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, DOI 10.1016/S0024-3795(02)00347-6
- Karlheinz Gröchenig, Localized frames are finite unions of Riesz sequences, Adv. Comput. Math. 18 (2003), no. 2-4, 149–157. Frames. MR 1968117, DOI 10.1023/A:1021368609918
- Richard V. Kadison and I. M. Singer, Extensions of pure states, Amer. J. Math. 81 (1959), 383–400. MR 123922, DOI 10.2307/2372748
- Joram Lindenstrauss and Lior Tzafriri, Classical Banach spaces. I, Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 92, Springer-Verlag, Berlin-New York, 1977. Sequence spaces. MR 0500056, DOI 10.1007/978-3-642-66557-8
Additional Information
- Peter G. Casazza
- Affiliation: Department of Mathematics, University of Missouri, Columbia, Missouri 65211
- MR Author ID: 45945
- 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
- MR Author ID: 256874
- Email: janet@math.missouri.edu
- 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
- © Copyright 2008
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication. - Journal: Proc. Amer. Math. Soc. 136 (2008), 2043-2053
- MSC (2000): Primary 46C05, 42C15, 46L05
- DOI: https://doi.org/10.1090/S0002-9939-08-09264-2
- MathSciNet review: 2383510