Factorization of Reenement Masks of Function Vectors

Considering the set of closed shift{invariant subspaces V j (j 2 Z Z) of L 2 (l R) generated by a reenable function vector , we give necessary and suucient conditions for the reenement mask of ensuring controlled approximation order m. In particular, algebraic poly-nomials can be exactly reproduced in V 0 if and only if the reenement mask of can be factorized. The results are illustrated by B{splines with multiple knots. x1 Introduction The idea of considering a ladder of imbedded subspaces V j of a Hilbert space for approximating functions has extensively been used in many applications. In the case of multiresolution analysis of L 2 (l R), the subspaces V j are usually generated by a single function 2 L 2 (l R), V j := clos L 2 span f(2 j ?l) : l 2 Z Zg: In order to ensure the condition V j V j+1 (j 2 Z Z), we need a reenable scaling function, i.e., has to satisfy a functional equation of the type = X l2Z Z p l (2 ?l) (fp l g l2Z Z 2 l 2): (1:1) A lot of papers have been delt with solutions of (1.1) and with their properties. Functions satisfying (1.1) not only arise in the context of multiresolu-tion analysis and wavelets. They also play an important role in subdivision schemes. By convenient choice of the reenement mask P := X l2Z Z p l e ?il 1 pp. 1{8. 2 G. Plonka one is able to innuence properties of like regularity or support and, at the same time, properties of the generated set of subspaces V j of L 2 (l R). The close connection between the reenement mask of and the structure of the shift{invariant subspaces generated by is the clue for many successful applications of the theory of multiresolution and corresponding wavelets. For example, V 0 provides controlled approximation order m if and only if the reenement mask P factorizes P(u) = 1 + e ?iu 2 m S(u) (1:2) with an appropriate chosen 2-periodic function S (cf. 2{4]). In the last time, also the generalized multiresolution analysis of multici-plicity r (r 2 l N) of L 2 (l R) has been considered in more detail (cf. 5{8]). Now the set of imbedded closed subspaces V j of L 2 (l R) is generated by a function vector := () r?1 =0 …