Projective multiresolution analyses for L²(R²)

2003/08/31 by Judith A. Packer, Marc A. Rieffel
Mathematics · #math.FA #math.OA #msc:46L99 #msc:42C15 #msc:46H25 #msc:47A05

paper · pdf

published as J. Fourier Anal. Appl. 10 (2004), no. 5, 439--464. · 25 pages

arxiv created 2004/03/09 · arxiv updated 2009/12/01

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 L²(\brn). Then we show how to construct examples for the case of any diagonal 2 × 2 dilation matrix with integer entries, with initial module specified to be any fixed finitely generated projective C(\mathbb T²)-module. We compute the isomorphism classes of the corresponding wavelet modules.

Related

• Projective multiresolution analyses for dilations in higher dimensions
• Projective multi-resolution analyses arising from direct limits of Hilbert modules
• Symmetries in projective multiresolution analyses
• Wavelet filter functions, the matrix completion problem, and projective modules over C(\mathbb Tⁿ)
• Spectral Models for Orthonormal Wavelets and Multiresolution Analysis of L²(\mathbb R)