Classification of Generalized Multiresolution Analyses

Details Classification of Generalized Multiresolution Analyses Classification of Generalized Multiresolution Analyses

2009-10-28

arxiv.org/abs/0910.5446v1

Mathematics

Classical Analysis and ODEs

18 pages including bibliograph

Scientific paper

We discuss how generalized multiresolution analyses (GMRAs), both classical and those defined on abstract Hilbert spaces, can be classified by their multiplicity functions $m$ and matrix-valued filter functions $H$. Given a natural number valued function $m$ and a system of functions encoded in a matrix $H$ satisfying certain conditions, a construction procedure is described that produces an abstract GMRA with multiplicity function $m $ and filter system $H$. An equivalence relation on GMRAs is defined and described in terms of their associated pairs $(m,H)$. This classification system is applied to classical examples in $L^2 (\mathbb R^d)$ as well as to previously studied abstract examples.

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