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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

The noncommutative Wiener lemma, linear independence, and spectral properties of the algebra of time-frequency shift operators


Author: Radu Balan
Journal: Trans. Amer. Math. Soc. 360 (2008), 3921-3941
MSC (2000): Primary 43A20; Secondary 42C15, 46H30
Published electronically: January 11, 2008
MathSciNet review: 2386252
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Abstract: In this paper we analyze the Banach *-algebra of time-frequency shifts with absolutely summable coefficients. We prove a noncommutative version of the Wiener lemma. We also construct a faithful tracial state on this algebra which proves the algebra contains no compact operators. As a corollary we obtain a special case of the Heil-Ramanathan-Topiwala conjecture regarding linear independence of finitely many time-frequency shifts of one $ L^2$ function. We also estimate the coefficient decay of the inverse of finite linear combinations of time-frequency shifts.


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Additional Information

Radu Balan
Affiliation: Siemens Corporate Research, 755 College Road East, Princeton, New Jersey 08540
Address at time of publication: Department of Mathematics, University of Maryland, College Park, Maryland 20742
Email: radu.balan@siemens.com, rvbalan@math.umd.edu

DOI: http://dx.doi.org/10.1090/S0002-9947-08-04448-6
PII: S 0002-9947(08)04448-6
Keywords: Time-frequency shifts, operator algebras, Wiener lemma, trace
Received by editor(s): November 9, 2005
Received by editor(s) in revised form: October 3, 2006
Published electronically: January 11, 2008
Article copyright: © Copyright 2008 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.