Generalized multiresolution analyses with given multiplicity functions

Details Generalized multiresolution analyses with given multiplicity functions Generalized multiresolution analyses with given multiplicity functions

2007-10-10

arxiv.org/abs/0710.2071v1

Mathematics

Functional Analysis

16 pages including bibliography

Scientific paper

Generalized multiresolution analyses are increasing sequences of subspaces of a Hilbert space $\H$ that fail to be multiresolution analyses in the sense of wavelet theory because the core subspace does not have an orthonormal basis generated by a fixed scaling function. Previous authors have studied a multiplicity function $m$ which, loosely speaking, measures the failure of the GMRA to be an MRA. When the Hilbert space $\H$ is $L^2(\mathbb R^n)$, the possible multiplicity functions have been characterized by Baggett and Merrill. Here we start with a function $m$ satisfying a consistency condition which is known to be necessary, and build a GMRA in an abstract Hilbert space with multiplicity function $m$.

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