Generalized multiresolution analyses with given multiplicity functions

TL;DR

This paper constructs generalized multiresolution analyses (GMRAs) in abstract Hilbert spaces based on a given multiplicity function, extending wavelet theory beyond classical MRA limitations.

Contribution

It provides a method to build GMRAs from a specified multiplicity function satisfying a known consistency condition, generalizing previous characterizations.

Findings

Constructed GMRAs from multiplicity functions in abstract Hilbert spaces.

Extended wavelet theory to non-orthonormal core subspaces.

Connected multiplicity functions with GMRA existence.

Abstract

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 $m$.