ACCURACY OF LATTICE TRANSLATES OF SEVERAL MULTIDIMENSIONAL REFINABLE FUNCTIONS

Complex-valued functions f(1),..., f(r) on R-d are refinable if they a re linear combinations of finitely many of the rescaled and translated functions f(i)(Ax - k), where the translates k are taken along a latt ice Gamma subset of R-d and A is a dilation matrix that expansively ma ps Gamma into itself. Refinable functions satisfy a refinement equatio n f(x) = Sigma(k is an element of Lambda)c(k)(Ax - k), where Lambda is a finite subset of Gamma, the c(k) are r x r matrices, and f(x) = (f( 1)(x), ..., f(r)(x))(T). The accuracy of f is the highest degree p suc h that all multivariate polynomials q with degree(q) < p are exactly r eproduced from linear combinations of translates of f(1,) ..., f(r) al ong the lattice Gamma. In this paper, we determine the accuracy p from the matrices c(k). Moreover, we determine explicitly the coefficients gamma(alpha,i)(k) such that x(alpha) = Sigma(i=1)(r)Sigma(k is an ele ment of Gamma)gamma(alpha,i)(k)f(i)(x + k). These coefficients are mul tivariate polynomials gamma(alpha,i)(x) Of degree \alpha\ evaluated at lattice points k is an element of Gamma. (C) 1998 Academic Press.