2001/07/31 by Judith A. Packer, Marc A. Rieffel
Mathematics · #math.FA #math.CA #math.OA #msc:46L99 #msc:42C40 #msc:46H25
published as J. Fourier Anal. Appl. 9 (2003), no. 2, 101--116 · 21 pages, various local improvements
arxiv created 2002/03/18 · arxiv updated 2009/11/30
We discuss how one can use certain filters from signal processing to describe isomorphisms between certain projective C(\mathbb Tn)-modules. Conversely, we show how cancellation properties for finitely generated projective modules over C(\mathbb Tn) 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.