APPROXIMATION BY MULTIPLE REFINABLE FUNCTIONS

We consider the shift-invariant space, S(Phi), generated by a set Phi = {phi(1),...,phi(r)} of compactly supported distributions on R when t he vector of distributions phi := (phi(1),..., phi(r))(T) satisfies a system of refinement equations expressed in matrix form as phi = Sigma (alpha is an element of Z) a(alpha)phi(2 . - alpha) where a is a finit ely supported sequence of r x r matrices of complex numbers. Such mult iple refinable functions occur naturally in the study of multiple wave lets. The purpose of the present paper is to characterize the accuracy of Phi, the order of the polynomial space contained in S(Phi), strict ly in terms of the refinement mask a. The accuracy determines the L-p- approximation order of S(Phi) when the functions in Phi belong to L-p( R) (see Jia [10]). The characterization is achieved in terms of the ei genvalues and eigenvectors of the subdivision operator associated with the mask a. In particular, they extend and improve the results of Hei l, Strang and Strela [7], and of Plonka [16]. In addition, a counterex ample is given to the statement of Strang and Strela [20] that the eig envalues of the subdivision operator determine the accuracy. The resul ts do not require the linear independence of the shifts of phi.