Factorization of re nement masks of function vectors

Considering the set of closed shift invariant subspaces Vj j Z of L lR generated by a re nable function vector we give necessary and su cient conditions for the re nement mask of ensur ing controlled approximation order m In particular algebraic poly nomials can be exactly reproduced in V if and only if the re nement mask of can be factorized The results are illustrated by B splines with multiple knots x Introduction The idea of considering a ladder of imbedded subspaces Vj of a Hilbert space for approximating functions has extensively been used in many ap plications In the case of multiresolution analysis of L lR the subspaces Vj are usually generated by a single function L lR Vj closL span f j l l Zg In order to ensure the condition Vj Vj j Z we need a re nable scaling function i e has to satisfy a functional equation of the type X