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 lowpass filter. However, we also give an example of a continuous lowpass filter for which it is impossible to find corresponding continuous highpass 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 eprints
 Pub Date:
 July 2001
 arXiv:
 arXiv:math/0107231
 Bibcode:
 2001math......7231P
 Keywords:

 Mathematics  Functional Analysis;
 Mathematics  Classical Analysis and ODEs;
 Mathematics  Operator Algebras;
 46L99 (primary);
 42C40;
 46H25 (secondary)
 EPrint:
 21 pages, various local improvements