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

Judith A. Packer·Marc A. Rieffel
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Abstract

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 $C(\btn)$ 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 $L^2(\brn)$. 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.

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Judith A. Packer, Marc A. Rieffel. 2004-03-09. Projective multiresolution analyses for $L^2(R^2)$. https://arxiv.org/abs/math/0308132

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