Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)



Projections in noncommutative tori and Gabor frames

Author: Franz Luef
Journal: Proc. Amer. Math. Soc. 139 (2011), 571-582
MSC (2010): Primary 42C15, 46L08; Secondary 22D25, 43A20
Published electronically: July 16, 2010
MathSciNet review: 2736339
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: We describe a connection between two seemingly different problems: (a) the construction of projections in noncommutative tori and (b) the construction of tight Gabor frames for $ L^2(\mathbb{R})$. The present investigation relies on interpretation of projective modules over noncommutative tori in terms of Gabor analysis. The main result demonstrates that Rieffel's condition on the existence of projections in noncommutative tori is equivalent to the Wexler-Raz biorthogonality relations for tight Gabor frames. Therefore we are able to invoke results on the existence of Gabor frames in the construction of projections in noncommutative tori. In particular, the projection associated with a Gabor frame generated by a Gaussian turns out to be Boca's projection. Our approach to Boca's projection allows us to characterize the range of existence of Boca's projection. The presentation of our main result provides a natural approach to the Wexler-Raz biorthogonality relations in terms of Hilbert $ C^*$-modules over noncommutative tori.

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

  • 1. F. Boca.
    Projections in rotation algebras and theta functions.
    Comm. Math. Phys., 202(2):325-357, 1999. MR 1690050 (2000j:46101)
  • 2. A. Connes.
    $ C^*$-algébres et géométrie différentielle.
    C. R. Acad. Sci. Paris Sér. A-B, 290(13):A599-A604, 1980. MR 572645 (81c:46053)
  • 3. A. Connes.
    An analogue of the Thom isomorphism for crossed products of a $ C^*$-algebra by an action of $ \mathbb{R}$.
    Adv. in Math., 39(1):31-55, 1981. MR 605351 (82j:46084)
  • 4. A. Connes.
    Noncommutative geometry.
    Academic Press Inc., San Diego, CA, 1994. MR 1303779 (95j:46063)
  • 5. I. Daubechies, H. J. Landau, and Z. Landau.
    Gabor time-frequency lattices and the Wexler-Raz identity.
    J. Fourier Anal. Appl., 1(4):437-478, 1995. MR 1350701 (96i:42021)
  • 6. G. A. Elliott and Q. Lin.
    Cut-down method in the inductive limit decomposition of non-commutative tori.
    J. London Math. Soc. (2), 54(1):121-134, 1996. MR 1395072 (98e:46067)
  • 7. H. G. Feichtinger.
    On a new Segal algebra.
    Monatsh. Math., 92:269-289, 1981. MR 643206 (83a:43002)
  • 8. H. G. Feichtinger.
    Modulation spaces on locally compact Abelian groups.
    Technical report, January 1983. Also in
    R. Radha, M. Krishna, and S. Thangavelu, editors, Proc. Internat. Conf. on Wavelets and Applications, pages 1-56, Chennai, January 2002, 2003. New Delhi Allied Publishers.
  • 9. H. G. Feichtinger.
    Modulation Spaces: Looking Back and Ahead.
    Sampl. Theory Signal Image Process., 5(2):109-140, 2006.
  • 10. H. G. Feichtinger and W. Kozek.
    Quantization of TF lattice-invariant operators on elementary LCA groups.
    In H. Feichtinger and T. Strohmer, editors, Gabor Analysis and Algorithms. Theory and Applications, Applied and Numerical Harmonic Analysis, pages 233-266, 452-488, Birkhäuser Boston, Boston, MA, 1998. MR 1601091 (98j:42019); MR 1601119 (98h:42001)
  • 11. H. G. Feichtinger and F. Luef.
    Wiener Amalgam Spaces for the Fundamental Identity of Gabor Analysis.
    Collect. Math., 57(Extra Volume (2006)):233-253, 2006.
  • 12. D. Gabor.
    Theory of communication.
    J. IEEE, 93(26):429-457, 1946.
  • 13. K. Gröchenig.
    Foundations of Time-Frequency Analysis.
    Appl. Numer. Harmon. Anal. Birkhäuser Boston, Boston, MA, 2001. MR 1843717 (2002h:42001)
  • 14. K. Gröchenig and M. Leinert.
    Wiener's Lemma for Twisted Convolution and Gabor Frames.
    J. Amer. Math. Soc., 17:1-18, 2004. MR 2015328 (2004m:42037)
  • 15. K. Gröchenig.
    Weight functions in time-frequency analysis.
    In L. Rodino and et al., editors, Pseudodifferential Operators: Partial Differential Equations and Time-Frequency Analysis, volume 52, 343 - 366, 2007. MR 2385335 (2009c:42070)
  • 16. A. J. E. M. Janssen.
    Duality and biorthogonality for Weyl-Heisenberg frames.
    J. Fourier Anal. Appl., 1(4):403-436, 1995. MR 1350700 (97e:42007)
  • 17. A. J. E. M. Janssen and T. Strohmer.
    Characterization and computation of canonical tight windows for Gabor frames.
    J. Fourier Anal. Appl., 8(1):1-28, 2002. MR 1882363 (2002m:42040)
  • 18. A. J. E. M. Janssen and T. Strohmer.
    Hyperbolic secants yield Gabor frames.
    Appl. Comput. Harmon. Anal., 12(2):259-267, 2002. MR 1884237 (2002k:42073)
  • 19. A. J. E. M. Janssen.
    On generating tight Gabor frames at critical density.
    J. Fourier Anal. Appl., 9(2):175-214, 2003. MR 1964306 (2003m:42054)
  • 20. F. Luef.
    On spectral invariance of non-commutative tori.
    In Operator Theory, Operator Algebras, and Applications, volume 414, pages 131-146. American Mathematical Society, 2006. MR 2277208 (2008k:46211)
  • 21. F. Luef.
    Gabor analysis, noncommutative tori and Feichtinger's algebra.
    In Gabor and Wavelet Frames, volume 10 of IMS Lecture Notes Series. World Sci. Pub., 2007. MR 2428027 (2010a:46172)
  • 22. F. Luef.
    Projective modules over noncommutative tori are multi-window Gabor frames for modulation spaces.
    J. Funct. Anal., 257(6):1921-1946, 2009. MR 2540994
  • 23. F. Luef and Y. Manin.
    Quantum theta functions and Gabor frames for modulation spaces.
    Lett. Math. Phys., 88(1-3):131-161, 2009. MR 2512143
  • 24. Y. I. Lyubarskij.
    Frames in the Bargmann space of entire functions.
    In Entire and subharmonic functions, volume 11 of Adv. Sov. Math., pages 167-180. American Mathematical Society (AMS), Providence, RI, 1992. MR 1188007 (93k:30036)
  • 25. Y. I. Manin.
    Theta functions, quantum tori and Heisenberg groups.
    Lett. Math. Phys., 56(3):295-320, 2001. MR 1855265 (2003a:14065)
  • 26. Y. I. Manin.
    Real multiplication and noncommutative geometry (ein Alterstraum).
    In The legacy of Niels Henrik Abel, pages 685-727. Springer, Berlin, 2004. MR 2077591 (2006e:11077)
  • 27. Y. I. Manin.
    Functional equations for quantum theta functions.
    Publ. Res. Inst. Math. Sci., 40(3):605-624, 2004. MR 2074694 (2006b:53114)
  • 28. M. Marcolli.
    Arithmetic Noncommutative Geometry.
    University Lecture Series, volume 36, AMS, 2005. MR 2159918 (2006g:58018)
  • 29. K. A. Okoudjou.
    Embedding of some classical Banach spaces into modulation spaces.
    Proc. Amer. Math. Soc., 132(6):1639-1647, 2004. MR 2051124 (2005b:46074)
  • 30. M. A. Rieffel.
    $ C$*-algebras associated with irrational rotations.
    Pac. J. Math., 93:415-429, 1981. MR 623572 (83b:46087)
  • 31. M. A. Rieffel.
    Projective modules over higher-dimensional noncommutative tori.
    Can. J. Math., 40(2):257-338, 1988. MR 941652 (89m:46110)
  • 32. A. Ron and Z. Shen.
    Weyl-Heisenberg frames and Riesz bases in $ {L}_2(\mathbb{R}^d)$.
    Duke Math. J., 89(2):237-282, 1997. MR 1460623 (98i:42013)
  • 33. K. Seip.
    Density theorems for sampling and interpolation in the Bargmann-Fock space. I.
    J. Reine Angew. Math., 429:91-106, 1992. MR 1173117 (93g:46026a)
  • 34. J. Wexler and S. Raz.
    Discrete Gabor expansions.
    Signal Proc., 21:207-220, 1990.

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (2010): 42C15, 46L08, 22D25, 43A20

Retrieve articles in all journals with MSC (2010): 42C15, 46L08, 22D25, 43A20

Additional Information

Franz Luef
Affiliation: Fakultät für Mathematik, Universität Wien, Nordbergstrasse 15, 1090 Wien, Austria
Address at time of publication: Department of Mathematics, University of California, Berkeley, California 94720-3840

Keywords: Gabor frames, noncommutative tori, projections in $C^{*}$-algebras
Received by editor(s): November 13, 2009
Received by editor(s) in revised form: March 10, 2010
Published electronically: July 16, 2010
Additional Notes: The author was supported by the Marie Curie Excellence grant MEXT-CT-2004-517154 and the Marie Curie Outgoing Fellowship PIOF-220464.
Communicated by: Marius Junge
Article copyright: © Copyright 2010 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

American Mathematical Society