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

 

Geometric aspects of frame representations of abelian groups


Authors: Akram Aldroubi, David Larson, Wai-Shing Tang and Eric Weber
Journal: Trans. Amer. Math. Soc. 356 (2004), 4767-4786
MSC (2000): Primary 43A70, 94A20, 42C40; Secondary 43A45, 46N99
Published electronically: June 22, 2004
MathSciNet review: 2084397
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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
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
Email: esweber@iastate.edu

DOI: http://dx.doi.org/10.1090/S0002-9947-04-03679-7
PII: S 0002-9947(04)03679-7
Keywords: Regular sampling, periodic sampling, multiplexing, locally compact abelian group, frame representation, spectral multiplicity, wavelet, Weyl-Heisenberg frame
Received by editor(s): March 4, 2002
Published electronically: June 22, 2004
Article copyright: © Copyright 2004 American Mathematical Society