An algebraic approach to multiresolution analysis

Foote, Richard

doi:10.1090/S0002-9947-05-03656-1

An algebraic approach to multiresolution analysis
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by Richard Foote PDF

Trans. Amer. Math. Soc. 357 (2005), 5031-5050 Request permission

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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