Multiple refinable Hermite interpolants

where phi = (phi(1),..., phi(r))(T) is a vector of compactly supported func tions on R and a is a finitely supported sequence of r x r matrices called the refinement mask. If phi is a continuous solution and a is supported on [N-1, N-2], then v := (phi(n))(n=N1)(N2-1) is an eigenvector of the matrix (a(2k - n))(k, n = N1)(N2 - 1) associated with eigenvalue 1. Conversely, gi ven such an eigenvector v, we may ask whether there exists a continuous sol ution phi such that phi(n) = v(n) for N-1 less than or equal to n less than or equal to N-2 - 1 (phi(n) = 0 for n is not an element of [N-1 . N2 - 1]. according to the support ). The first part of this paper answers this ques tion completely. This existence problem is more general than either the con vergence of the subdivision scheme or the requirement of stability, since i n one of the latter cases, the eigenvector v, is unique up to a constant mu ltiplication. The second part of this paper is concerned with Hermite inter polant solutions, i.e., fur some n(0) epsilon Z and j m = 1,..., r, phi(f) epsilon Cr-1(R) and phi(f)((m-1)) (n) = delta(j, m)delta(n, n0), n epsilon Z. We provide a necessary and sufficient condition for the refinement equat ion to have an Hermite interpolant solution. The condition is strictly in t erms of the refinement mask. Our method is to characterize the existence an d the Hermite interpolant condition by joint spectral radii of matrices. Se veral concrete examples are presented to illustrate the general theory. (C) 2000 Academic Press.