Spanning and independence properties of frame partitions
HTML articles powered by AMS MathViewer
- by Bernhard G. Bodmann, Peter G. Casazza, Vern I. Paulsen and Darrin Speegle PDF
- Proc. Amer. Math. Soc. 140 (2012), 2193-2207 Request permission
Abstract:
We answer a number of open problems in frame theory concerning the decomposition of frames into linearly independent and/or spanning sets. We prove that Parseval frames with norms bounded away from $1$ can be decomposed into a number of sets whose complements are spanning, where the number of these sets only depends on the norm bound. Further, we prove a stronger result for Parseval frames whose norms are uniformly small, which shows that in addition to the spanning property, the sets can be chosen to be independent and the complement of each set can contain a number of disjoint, spanning sets.References
- Joel Anderson, Extreme points in sets of positive linear maps on ${\cal B}({\cal H})$, J. Functional Analysis 31 (1979), no. 2, 195â217. MR 525951, DOI 10.1016/0022-1236(79)90061-2
- Bernhard G. Bodmann, Peter G. Casazza, and Gitta Kutyniok, A quantitative notion of redundancy for finite frames, Appl. Comput. Harmon. Anal. 30 (2011), no. 3, 348â362. MR 2784569, DOI 10.1016/j.acha.2010.09.004
- Peter G. Casazza, Ole Christensen, Alexander M. Lindner, and Roman Vershynin, Frames and the Feichtinger conjecture, Proc. Amer. Math. Soc. 133 (2005), no. 4, 1025â1033. MR 2117203, DOI 10.1090/S0002-9939-04-07594-X
- 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. 299â355. MR 2277219, DOI 10.1090/conm/414/07820
- Peter G. Casazza, Gitta Kutyniok, and Darrin Speegle, A redundant version of the Rado-Horn theorem, Linear Algebra Appl. 418 (2006), no. 1, 1â10. MR 2257571, DOI 10.1016/j.laa.2006.01.010
- Peter G. Casazza, Gitta Kutyniok, Darrin Speegle, and Janet C. Tremain, A decomposition theorem for frames and the Feichtinger conjecture, Proc. Amer. Math. Soc. 136 (2008), no. 6, 2043â2053. MR 2383510, DOI 10.1090/S0002-9939-08-09264-2
- Peter G. Casazza and Janet Crandell Tremain, The Kadison-Singer problem in mathematics and engineering, Proc. Natl. Acad. Sci. USA 103 (2006), no. 7, 2032â2039. MR 2204073, DOI 10.1073/pnas.0507888103
- Ole Christensen, An introduction to frames and Riesz bases, Applied and Numerical Harmonic Analysis, BirkhÀuser Boston, Inc., Boston, MA, 2003. MR 1946982, DOI 10.1007/978-0-8176-8224-8
- Ingrid Daubechies, From the original framer to present-day time-frequency and time-scale frames, J. Fourier Anal. Appl. 3 (1997), no. 5, 485â486. Dedicated to the memory of Richard J. Duffin. MR 1491928, DOI 10.1007/BF02648878
- Ingrid Daubechies and Bin Han, The canonical dual frame of a wavelet frame, Appl. Comput. Harmon. Anal. 12 (2002), no. 3, 269â285. MR 1912147, DOI 10.1006/acha.2002.0381
- Ingrid Daubechies, Bin Han, Amos Ron, and Zuowei Shen, Framelets: MRA-based constructions of wavelet frames, Appl. Comput. Harmon. Anal. 14 (2003), no. 1, 1â46. MR 1971300, DOI 10.1016/S1063-5203(02)00511-0
- Jack Edmonds and D. R. Fulkerson, Transversals and matroid partition, J. Res. Nat. Bur. Standards Sect. B 69B (1965), 147â153. MR 188090, DOI 10.6028/jres.069B.016
- Karlheinz Gröchenig, Foundations of time-frequency analysis, Applied and Numerical Harmonic Analysis, BirkhÀuser Boston, Inc., Boston, MA, 2001. MR 1843717, DOI 10.1007/978-1-4612-0003-1
- Alfred Horn, A characterization of unions of linearly independent sets, J. London Math. Soc. 30 (1955), 494â496. MR 71487, DOI 10.1112/jlms/s1-30.4.494
- J. KovaÄeviÄ and A. Chebira, Life beyond bases: The advent of frames (Part I), IEEE Signal Proc. Mag. 24(4) (2007), 86â104.
- J. KovaÄeviÄ and A. Chebira, Life beyond bases: The advent of frames (Part II), IEEE Signal Proc. Mag. 24(5) (2007), 115â125.
- James G. Oxley, Matroid theory, Oxford Science Publications, The Clarendon Press, Oxford University Press, New York, 1992. MR 1207587
- R. Rado, A combinatorial theorem on vector spaces, J. London Math. Soc. 37 (1962), 351â353. MR 146186, DOI 10.1112/jlms/s1-37.1.351
- Amos Ron and Zuowei Shen, Affine systems in $L_2(\mathbf R^d)$: the analysis of the analysis operator, J. Funct. Anal. 148 (1997), no. 2, 408â447. MR 1469348, DOI 10.1006/jfan.1996.3079
Additional Information
- Bernhard G. Bodmann
- Affiliation: Department of Mathematics, 651 Philip G. Hoffman Hall, University of Houston, Houston, Texas 77204-3008
- MR Author ID: 644711
- Email: bgb@math.uh.edu
- Peter G. Casazza
- Affiliation: Department of Mathematics, University of Missouri, Columbia, Missouri 65211
- MR Author ID: 45945
- Email: casazzap@missouri.edu
- Vern I. Paulsen
- Affiliation: Department of Mathematics, 651 Philip G. Hoffman Hall, University of Houston, Houston, Texas 77204-3008
- MR Author ID: 137010
- ORCID: 0000-0002-2361-852X
- Email: vern@math.uh.edu
- Darrin Speegle
- Affiliation: Department of Mathematics and Computer Science, Saint Louis University, St. Louis, Missouri 63103
- Email: speegled@slu.edu
- Received by editor(s): October 29, 2010
- Received by editor(s) in revised form: February 14, 2011
- Published electronically: October 24, 2011
- Additional Notes: The first author was supported by NSF grant DMS-0807399
The second author was supported by NSF 1008183: DTRA/NSF 1042701
The third author was supported by NSF DMS-0600191
The fourth author was supported by NSF DMS-0354957 - Communicated by: Thomas Schlumprecht
- © Copyright 2011
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication. - Journal: Proc. Amer. Math. Soc. 140 (2012), 2193-2207
- MSC (2010): Primary 15A03, 42C15
- DOI: https://doi.org/10.1090/S0002-9939-2011-11072-4
- MathSciNet review: 2898683