Projective multiresolution analyses for dilations in higher dimensions

TL;DR

This paper develops a framework for constructing projective multiresolution analyses and wavelet frames in higher dimensions using dilation matrices and projective modules over functions on tori, extending previous methods.

Contribution

It generalizes the construction of projective multiresolution analyses to higher dimensions and conjugate dilation matrices, providing new examples of non-free projective module wavelet frames.

Findings

Constructed projective multiresolution analyses for higher-dimensional dilations.

Extended results to conjugate dilation matrices via SL(n,Z) transformations.

Embedded finitely generated modules as initial modules in the case n=3.

Abstract

We continue the study of projective module wavelet frames corresponding to diagonal dilation matrices on $\mathbb R^n$ with integer entries, focusing on the construction of a projective multi-resolution analysis corresponding to dilations whose domains are finitely generated projective modules over continuous complex-valued functions on $n$-tori, $n\geq 3$. We are able to generalize some of these results to dilation matrices that are conjugates of integral diagonal dilation matrices by elements of $SL(n,\mathbb Z).$ We follow the method proposed by the author and M. Rieffel, and are able to come up with examples of non-free projective module wavelet frames which can be described via this construction. As an application of our results, in the case $n=3,$ when the dilation matrix is a constant multiple of the identity, we embed every finitely generated module as an initial module.