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

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

2001-07-31

arxiv.org/abs/math/0107231v2

J. Fourier Anal. Appl. 9 (2003), no. 2, 101--116

Mathematics

Functional Analysis

21 pages, various local improvements

Scientific paper

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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