Projective multiresolution analyses for dilations in higher dimensions

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.