Wavelet filter functions, the matrix completion problem, and projective modules over $C(\mathbb T^n)$
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.
- Publication:
-
arXiv Mathematics e-prints
- Pub Date:
- July 2001
- DOI:
- 10.48550/arXiv.math/0107231
- arXiv:
- arXiv:math/0107231
- Bibcode:
- 2001math......7231P
- Keywords:
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- Mathematics - Functional Analysis;
- Mathematics - Classical Analysis and ODEs;
- Mathematics - Operator Algebras;
- 46L99 (primary);
- 42C40;
- 46H25 (secondary)
- E-Print:
- 21 pages, various local improvements