Wavelet Filter Functions, the Matrix Completion Problem, and Projective Modules over C(t N )

We discuss how one can use certain filters from signal processing to describe isomorphisms between certain projective C(T)-modules. Conversely, we show how cancellation properties for finitely generated projective modules over C(T) can often be used to prove the existence of continuous high pass filters, of the kind needed for multi-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. A key technique in wavelet theory is to use suitable low-pass filters to construct scaling functions and their multi-resolution analyses, and then to use corresponding high-pass filters to construct the corresponding wavelets. Thus once one has used a low-pass filter to construct a scaling function, it is important to be able to find associated high-pass filters. (They are not unique.) It has been pointed out several times in the literature that, given a continuous low-pass filter for dilation by q > 2, there is in general no continuous selection function for the construction of associated continuous high-pass filters, because of the existence of topological obstructions. (See the substantial discussion in section 2 of [6], and the references cited there.) It is known [16], [34], [6] that associated high-pass filters which are measurable will always exist. (See also the general Hilbert-space treatment of the existence of wavelets given in [2].) The problem of finding appropriate high-pass 1991 Mathematics Subject Classification. Primary 46L99; Secondary 42C40, 46H25.