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An algebraic approach to multiresolution analysis


Author: Richard Foote
Journal: Trans. Amer. Math. Soc. 357 (2005), 5031-5050
MSC (2000): Primary 20C99; Secondary 42C40
DOI: https://doi.org/10.1090/S0002-9947-05-03656-1
Published electronically: March 18, 2005
MathSciNet review: 2165396
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Abstract: The notion of a weak multiresolution analysis is defined over an arbitrary field in terms of cyclic modules for a certain affine group ring. In this setting the basic properties of weak multiresolution analyses are established, including characterizations of their submodules and quotient modules, the existence and uniqueness of reduced scaling equations, and the existence of wavelet bases. These results yield some standard facts on classical multiresolution analyses over the reals as special cases, but provide a different perspective by not relying on orthogonality or topology. Connections with other areas of algebra and possible further directions are mentioned.


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Additional Information

Richard Foote
Affiliation: Department of Mathematics and Statistics, University of Vermont, Burlington, Vermont 05405
Email: foote@math.uvm.edu

DOI: https://doi.org/10.1090/S0002-9947-05-03656-1
Keywords: Multiresolution analysis, wavelets
Received by editor(s): March 7, 2003
Received by editor(s) in revised form: January 14, 2004, and March 5, 2004
Published electronically: March 18, 2005
Additional Notes: This work was partially supported by an AFOSR/NM grant
Article copyright: © Copyright 2005 American Mathematical Society

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