Generalized Low Pass Filters and MRA Frame Wavelets

A tight frame wavelet psi is an L-2(R) function such that {psi(jk) (x)} = {2(J/2) psi(2(j) x -k), j, k is an element of Z}, is a tight frame for L-2(R). We introduce a class of "generalized low pass filters" that allows us to define (and construct) the subclass of MRA tight frame wavelets. This leads us to an associated class of "generalized scaling functions" that are not necessarily obtained from a multiresolution analysis. We study several properties of these classes of "generalized" wavelets, scaling functions and filters (such as their multipliers and their connectivity). We also compare our approach with those recently obtained by other authors.