2003/08/26 by Akram Aldroubi, Aldroubi, Akram, David R. Larson +6
Mathematics · #42C40 #43A70 #94A20 #Advanced Harmonic Analysis Research #FOS: Mathematics #Functional Analysis (math.FA) #Mathematical Analysis and Transform Methods #Mathematical functions and polynomials #math.FA #msc:42C40 #msc:43A70 #msc:94A20
paper · pdf · doi:10.48550/arxiv.math/0308250
20 pages; contact author: Eric Weber
arxiv created 2003/08/26 · openalex publication_date 2003/08/26 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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.