Wavelet filter functions, the matrix completion problem, and projective modules over C(double-struck T signn)

Abstract

We discuss how one can use certain filters from signal processing to describe isomorphisms between certain projective C(double-struck T signn)-modules. Conversely, we show how cancellation properties for finitely generated projective modules over C(double-struck T signn) 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.