Projective multiresolution analyses for $L^2(R^2)$

We define the notion of “projective” multiresolution analyses, for which, by definition, the initial space corresponds to a finitely generated projective module over the algebra $C(\text{btn})$ of continuous complex-valued functions on an $n$-torus. The case of ordinary multi-wavelets is that in which the projective module is actually free. We discuss the properties of projective multiresolution analyses, including the frames which they provide for $L^2(\text{brn})$. Then we show how to construct examples for the case of any diagonal $2\times 2$ dilation matrix with integer entries, with initial module specified to be any fixed finitely generated projective $C(\mathbb T^2)$-module. We compute the isomorphism classes of the corresponding wavelet modules.