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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
MathSciNet review:
2383510
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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.
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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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