A Note on Construction of Biorthogonal Multi - Scaling Functions

In many applications biorthogonal wavelets prove to be more ee-cient than orthogonal ones. In this note we present a procedure for constructing biorthogonal multi-scaling functions with any given approximation order, starting from a low pass multi-lter H 0. We show that dual low pass multi-lter F 0 can be found if det(H 0 (z)) and det(H 0 (?z)) do not have common roots. We also suggest \two-scale balancing" as a way to enhance approximation properties of F 0 (z). As an illustration to our technique we construct multi-scaling functions with various smoothness biorthogonal to Hermite cu-bics and a pair of low-pass biorthogonal multi-lters based on the well known GHM orthogonal example.