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

 

Varying the time-frequency lattice of Gabor frames


Authors: Hans G. Feichtinger and Norbert Kaiblinger
Journal: Trans. Amer. Math. Soc. 356 (2004), 2001-2023
MSC (2000): Primary 42C15; Secondary 47B38, 81R30, 94A12
Published electronically: November 12, 2003
MathSciNet review: 2031050
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Abstract: A Gabor or Weyl-Heisenberg frame for $L^2(\mathbb{R}^d)$is generated by time-frequency shifts of a square-integrable function, the Gabor atom, along a time-frequency lattice. The dual frame is again a Gabor frame, generated by the dual atom. In general, Gabor frames are not stable under a perturbation of the lattice constants; that is, even for arbitrarily small changes of the parameters the frame property can be lost.

In contrast, as a main result we show that this kind of stability does hold for Gabor frames generated by a Gabor atom from the modulation space $M^1(\mathbb{R}^d)$, which is a dense subspace of $L^2(\mathbb{R}^d)$. Moreover, in this case the dual atom depends continuously on the lattice constants. In fact, we prove these results for more general weighted modulation spaces. As a consequence, we obtain for Gabor atoms from the Schwartz class that the continuous dependence of the dual atom holds even in the Schwartz topology. Also, we complement these main results by corresponding statements for Gabor Riesz sequences and their biorthogonal system.


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Additional Information

Hans G. Feichtinger
Affiliation: Department of Mathematics, University of Vienna, Strudlhofgasse 4, 1090 Vienna, Austria
Email: hans.feichtinger@univie.ac.at

Norbert Kaiblinger
Affiliation: Department of Mathematics, University of Vienna, Strudlhofgasse 4, 1090 Vienna, Austria
Address at time of publication: Georgia Institute of Technology, School of Mathematics, Atlanta, Georgia 30332-0160
Email: norbert.kaiblinger@univie.ac.at

DOI: http://dx.doi.org/10.1090/S0002-9947-03-03377-4
PII: S 0002-9947(03)03377-4
Keywords: Gabor frame, Weyl-Heisenberg frame, dual atom, Riesz basis, stability, perturbation, time-frequency lattice, modulation space, twisted convolution, coherent states
Received by editor(s): April 29, 2002
Received by editor(s) in revised form: April 9, 2003
Published electronically: November 12, 2003
Additional Notes: The second author was supported by the Austrian Science Fund FWF, grants P-14485 and J-2205
Article copyright: © Copyright 2003 American Mathematical Society