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)



A note on the density theorem for projective unitary representations

Author: Deguang Han
Journal: Proc. Amer. Math. Soc. 145 (2017), 1739-1745
MSC (2010): Primary 46C05, 46L10, 47D03
Published electronically: October 26, 2016
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: It is well known that a Gabor representation on $ L^{2}(\mathbb{R}^{d})$ admits a frame generator $ h\in L^{2}(\mathbb{R}^{d})$ if and only if the associated lattice satisfies the Beurling density condition, which in turn can be characterized as the ``trace condition'' for the associated von Neumann algebra. It happens that this trace condition is also necessary for any projective unitary representation of a countable group to admit a frame vector. However, it is no longer sufficient for general representations, and in particular not sufficient for Gabor representations when they are restricted to proper time-frequency invariant subspaces. In this short note we show that the condition is also sufficient for a large class of projective unitary representations, which implies that the Gabor density theorem is valid for subspace representations in the case of irrational types of lattices.

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

  • [1] Lawrence W. Baggett, Processing a radar signal and representations of the discrete Heisenberg group, Colloq. Math. 60/61 (1990), no. 1, 195-203. MR 1096368
  • [2] Bachir Bekka, Square integrable representations, von Neumann algebras and an application to Gabor analysis, J. Fourier Anal. Appl. 10 (2004), no. 4, 325-349. MR 2078261,
  • [3] Ingrid Daubechies, The wavelet transform, time-frequency localization and signal analysis, IEEE Trans. Inform. Theory 36 (1990), no. 5, 961-1005. MR 1066587,
  • [4] R. J. Duffin and A. C. Schaeffer, A class of nonharmonic Fourier series, Trans. Amer. Math. Soc. 72 (1952), 341-366. MR 0047179
  • [5] Jean-Pierre Gabardo and Deguang Han, Subspace Weyl-Heisenberg frames, J. Fourier Anal. Appl. 7 (2001), no. 4, 419-433. MR 1836821,
  • [6] Jean-Pierre Gabardo and Deguang Han, Frame representations for group-like unitary operator systems, J. Operator Theory 49 (2003), no. 2, 223-244. MR 1991737
  • [7] Jean-Pierre Gabardo, Deguang Han, and Yun-Zhang Li, Lattice tiling and density conditions for subspace Gabor frames, J. Funct. Anal. 265 (2013), no. 7, 1170-1189. MR 3073252,
  • [8] D. Gabor, Theory of communications, J. Inst. Elec. Eng. (London) 93 (1946), 429-457.
  • [9] Karlheinz Gröchenig, Foundations of time-frequency analysis, Applied and Numerical Harmonic Analysis, Birkhäuser Boston, Inc., Boston, MA, 2001. MR 1843717
  • [10] Deguang Han and David R. Larson, Frames, bases and group representations, Mem. Amer. Math. Soc. 147 (2000), no. 697, x+94. MR 1686653,
  • [11] Deguang Han and Yang Wang, Lattice tiling and the Weyl-Heisenberg frames, Geom. Funct. Anal. 11 (2001), no. 4, 742-758. MR 1866800,
  • [12] Deguang Han and Yang Wang, The existence of Gabor bases and frames, Wavelets, frames and operator theory, Contemp. Math., vol. 345, Amer. Math. Soc., Providence, RI, 2004, pp. 183-192. MR 2066828,
  • [13] Christopher Heil, History and evolution of the density theorem for Gabor frames, J. Fourier Anal. Appl. 13 (2007), no. 2, 113-166. MR 2313431,
  • [14] Christopher Heil, The density theorem and the homogeneous approximation property for Gabor frames, Representations, wavelets, and frames, Appl. Numer. Harmon. Anal., Birkhäuser Boston, Boston, MA, 2008, pp. 71-102. MR 2459314,
  • [15] John von Neumann, Mathematical foundations of quantum mechanics, translated by Robert T. Beyer, Princeton University Press, Princeton, 1955. MR 0066944
  • [16] Marc A. Rieffel, von Neumann algebras associated with pairs of lattices in Lie groups, Math. Ann. 257 (1981), no. 4, 403-418. MR 639575,

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (2010): 46C05, 46L10, 47D03

Retrieve articles in all journals with MSC (2010): 46C05, 46L10, 47D03

Additional Information

Deguang Han
Affiliation: Department of Mathematics, University of Central Florida, Orlando, Florida 32816

Keywords: Gabor frames, projective unitary representations, time-frequency lattice, von Neumann algebras.
Received by editor(s): April 22, 2016
Received by editor(s) in revised form: June 20, 2016, and June 21, 2016
Published electronically: October 26, 2016
Additional Notes: The author was partially supported by the NSF grants DMS-1106934 and DMS-1403400
Communicated by: Thomas Schlumprecht
Article copyright: © Copyright 2016 American Mathematical Society

American Mathematical Society