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

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

2003-08-14

arxiv.org/abs/math/0308132v2

J. Fourier Anal. Appl. 10 (2004), no. 5, 439--464.

Mathematics

Functional Analysis

25 pages

Scientific paper

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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