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Wavelet filter functions, the matrix completion problem, and projective modules over $C(\mathbb T^n)$

Judith A. Packer·Marc A. Rieffel
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Abstract

We discuss how one can use certain filters from signal processing to describe isomorphisms between certain projective $C(\mathbb T^n)$-modules. Conversely, we show how cancellation properties for finitely generated projective modules over $C(\mathbb T^n)$ can often be used to prove the existence of continuous high pass filters, of the kind needed for multivariate wavelets, corresponding to a given continuous low-pass filter. However, we also give an example of a continuous low-pass filter for which it is impossible to find corresponding continuous high-pass filters. In this way we give another approach to the solution of the matrix completion problem for filters of the kind arising in wavelet theory.

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Judith A. Packer, Marc A. Rieffel. 2002-03-18. Wavelet filter functions, the matrix completion problem, and projective modules over $C(\mathbb T^n)$. https://arxiv.org/abs/math/0107231

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