Remote Access Transactions of the American Mathematical Society
Green Open Access

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
DOI: https://doi.org/10.1090/S0002-9947-08-04448-6
Published electronically: January 11, 2008
MathSciNet review: 2386252
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

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.


References [Enhancements On Off] (What's this?)

  • 1. R. Balan, P. Casazza, C. Heil, and Z. Landau, Deficits and Excesses of Frames, Advances in Computational Mathematics 18 (2003), 93-116. MR 1968114 (2004a:42040)
  • 2. -, Excesses of Gabor Frames, Appl. Comput. Harmon. Anal. 14 (2003), 87-106. MR 1981203 (2004c:42058)
  • 3. -, Density, Overcompleteness, and Localization of Frames. I. Theory, J. Fourier Anal. Applic. 12 (2006), no. 2, 105-143. MR 2224392 (2007b:42041)
  • 4. -, Density, Overcompleteness, and Localization of Frames. II. Gabor Frames, J. Fourier Anal. Applic. 12 (2006), no. 3, 309-344. MR 2235170 (2007b:42042)
  • 5. A.G. Baskakov, Estimates for the entries of inverse matrices and the spectral analysis of linear operators, Izvestiya: Mathematics 61 (1997), no. 6, 1113-1135. MR 1609144 (98m:47007)
  • 6. I. Daubechies, H. Landau, and Z. Landau, Gabor Time-Frequency Lattices and the Wexler-Raz Identity, J. Fourier Analys. Applic. 1 (1995), no. 4, 437-478. MR 1350701 (96i:42021)
  • 7. M. Fornasier and K. Gröchenig, Intrinsic Localization of Frames, Constructive Approximation 22 (2005), no. 3, 395-415. MR 2164142 (2006f:42030)
  • 8. I. Gohberg, M.A. Kaashoek, and H.J. Woerdeman, The band method for positive and strictly contractive extension problems: an alternative version and new applications, Integral Equations and Operator Theory 12 (1989), 343-382. MR 998278 (90c:47022)
  • 9. K. Gröchenig, Foundations of Time-Frequency Analysis, Birkhäuser Boston, Inc., 2001.
  • 10. -, Localization of Frames, Banach Frames, and the Invertibility of Frame Operator, J. Fourier Anal. Applic. 10 (2004), 105-132. MR 2054304 (2005f:42086)
  • 11. -, Time-Frequency Analysis of Sjöstrand's Class, Revista Mat. Iberoam. 22 (2006), 703-724. MR 2294795
  • 12. K. Gröchenig and M. Leinert, Wiener's Lemma for Twisted Convolution and Gabor Frames, Journal of AMS 17 (2003), no. 1, 1-18.
  • 13. -, Symmetry of Matrix Algebras and Symbolic Calculus for Infinite Matrices, Trans. of AMS 358 (2006), no. 6, 2695-2711. MR 2204052 (2006k:47065)
  • 14. C. Heil, Linear Independence of Finite Gabor Systems, ``Harmonic Analysis and Applications", a volume in honor of John J. Benedetto's 65th birthday (C. Heil, ed.), Birkhäuser, 2005. MR 2249310 (2007d:42057)
  • 15. A. Hulanicki, On the spectral radius of hermitian elements in group algebras, Pacific J. Math. 18 (1966), 277-287. MR 0198267 (33:6426)
  • 16. -, On symmetry of group algebras of discrete nilpotent groups, Studia Math. 35 (1970), 207-219. MR 0278082 (43:3814)
  • 17. -, On the spectrum of convolution operators on groups with polynomial growth, Invent. Math. 17 (1972), 135-142. MR 0323951 (48:2304)
  • 18. A.J.E.M. Janssen, Duality and Biorthogonality for Weyl-Heisenberg Frames, J. Fourier Anal. Applic. 1 (1995), no. 4, 403-436. MR 1350700 (97e:42007)
  • 19. P.A. Linnell, Von Neumann algebras and linear independence of translates, Proc. AMS 127 (1999), 3269-3277. MR 1637388 (2000b:46106)
  • 20. L.H. Loomis, An Introduction to Abstract Harmonic Analysis, D. van Nostrand Company, Inc., 1953. MR 0054173 (14:883c)
  • 21. J. Ludwig, A class of symmetric and a class of Wiener group algebras, J. Funct. Anal. 31 (1979), no. 2, 187-194. MR 525950 (81a:43007)
  • 22. M.A. Neimark, Normed algebras, Wolters-Noordhoff Publishing, Groningen, third edition, 1972. MR 0438123 (55:11042)
  • 23. C.E. Rickart, General theory of Banach algebras, D. van Nostrand Co., Inc., 1960. MR 0115101 (22:5903)
  • 24. M.A. Rieffel, Von Neumann Algebras Associated with Pairs of Lattices in Lie Groups, Math. Anal. 257 (1981), 403-418. MR 639575 (84f:22010)
  • 25. F. Riesz and B.Sz. Nagy, Functional Analysis, Dover Publications, New York, 1990. MR 1068530 (91g:00002)
  • 26. L. Rodman, I.M. Spitkovsky, and H.J. Woerdeman, Carathéodory-Toeplitz and Nehari Problems for Matrix Valued Almost Periodic Functions, Trans. AMS 350 (1998), no. 6, 2185-2227. MR 1422908 (98h:47023)
  • 27. A. Ron and Z. Shen, Weyl-Heisenberg frames and Riesz bases in $ L_2(\bf R^d)$, Duke Math. J. 89 (1997), no. 2, 237-282. MR 1460623 (98i:42013)
  • 28. J. Sjöstrand, Wiener type algebras of pseudodifferential operators, Séminaire sur les Equations aux Dérivées Partielles, 1994-1995, École Polytech., Palaiseau 21 (1995), Exp. No. IV. MR 1362552 (96j:47049)
  • 29. T. Strohmer, Pseudodifferential operators and Banach algebras for mobile communications, Appl. Comp. Harmon. Anal. 20 (2006), no. 2, 237-249. MR 2207837
  • 30. -, personal communication (September 2005).
  • 31. A. Zygmund, Trigonometric Series, University Press, Cambridge, 3rd Edition, 2003.

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC (2000): 43A20, 42C15, 46H30

Retrieve articles in all journals with MSC (2000): 43A20, 42C15, 46H30


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: https://doi.org/10.1090/S0002-9947-08-04448-6
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.

American Mathematical Society