Skip to Main Content

Transactions of the American Mathematical Society

Published by the American Mathematical Society, the Transactions of the American Mathematical Society (TRAN) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6850 (online) ISSN 0002-9947 (print)

The 2020 MCQ for Transactions of the American Mathematical Society is 1.43.

What is MCQ? The Mathematical Citation Quotient (MCQ) measures journal impact by looking at citations over a five-year period. Subscribers to MathSciNet may click through for more detailed information.

 

Varying the time-frequency lattice of Gabor frames
HTML articles powered by AMS MathViewer

by Hans G. Feichtinger and Norbert Kaiblinger PDF
Trans. Amer. Math. Soc. 356 (2004), 2001-2023 Request permission

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.

References
Similar Articles
Additional Information
  • Hans G. Feichtinger
  • Affiliation: Department of Mathematics, University of Vienna, Strudlhofgasse 4, 1090 Vienna, Austria
  • MR Author ID: 65680
  • ORCID: 0000-0002-9927-0742
  • 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
  • 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
  • © Copyright 2003 American Mathematical Society
  • Journal: Trans. Amer. Math. Soc. 356 (2004), 2001-2023
  • MSC (2000): Primary 42C15; Secondary 47B38, 81R30, 94A12
  • DOI: https://doi.org/10.1090/S0002-9947-03-03377-4
  • MathSciNet review: 2031050