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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.

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Geometric aspects of frame representations of abelian groups
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by Akram Aldroubi, David Larson, Wai-Shing Tang and Eric Weber PDF
Trans. Amer. Math. Soc. 356 (2004), 4767-4786 Request permission

Abstract:

We consider frames arising from the action of a unitary representation of a discrete countable abelian group. We show that the range of the analysis operator can be determined by computing which characters appear in the representation. This allows one to compare the ranges of two such frames, which is useful for determining similarity and also for multiplexing schemes. Our results then partially extend to Bessel sequences arising from the action of the group. We apply the results to sampling on bandlimited functions and to wavelet and Weyl-Heisenberg frames. This yields a sufficient condition for two sampling transforms to have orthogonal ranges, and two analysis operators for wavelet and Weyl-Heisenberg frames to have orthogonal ranges. The sufficient condition is easy to compute in terms of the periodization of the Fourier transform of the frame generators.
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Additional Information
  • Akram Aldroubi
  • Affiliation: Department of Mathematics, Vanderbilt University, Nashville, Tennessee 37240
  • Email: aldroubi@math.vanderbilt.edu
  • David Larson
  • Affiliation: Department of Mathematics, Texas A&M University, College Station, Texas 77843-3368
  • MR Author ID: 110365
  • Email: larson@math.tamu.edu
  • Wai-Shing Tang
  • Affiliation: Department of Mathematics, National University of Singapore, 10 Kent Ridge Crescent, 119260, Republic of Singapore
  • Email: mattws@nus.edu.sg
  • Eric Weber
  • Affiliation: Department of Mathematics, Texas A&M University, College Station, Texas 77843-3368
  • Address at time of publication: Department of Mathematics, Iowa State University, 400 Carver Hall, Ames, Iowa 50011
  • MR Author ID: 660323
  • Email: esweber@iastate.edu
  • Received by editor(s): March 4, 2002
  • Published electronically: June 22, 2004
  • © Copyright 2004 American Mathematical Society
  • Journal: Trans. Amer. Math. Soc. 356 (2004), 4767-4786
  • MSC (2000): Primary 43A70, 94A20, 42C40; Secondary 43A45, 46N99
  • DOI: https://doi.org/10.1090/S0002-9947-04-03679-7
  • MathSciNet review: 2084397