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 | References | Similar Articles | Additional Information

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