Accuracy of several multidimensional refinable distributions

Compactly supported distributions f(1),..., f(r) on R-d are refinable if ea ch f(i) is a finite linear combination of the rescaled and translated distr ibutions f(j) (Ax - k), where the translates k are taken along a lattice Ga mma subset of R-d and A is a dilation matrix that expansively maps Gamma in to itself. Refinable distributions satisfy a refinement equation f (x) = Si gma(k is an element of Lambda) c(k) f (Ax - k), where Lambda is a finite su bset of Gamma, the c(k) are r x r matrices, and f = (f(1),...f(r))(T). The accuracy of f is the highest degree p such that all multivariate polynomial s q with degree(q) < p are exactly reproduced from linear combinations of t ranslates of f(1),...,f(r) along the lattice Gamma. We determine the accura cy p from the matrices c(k). Moreover we determine explicitly the coefficie nts y(alpha,i) (k) such that x(alpha) = Sigma(i=1)(r) Sigma(k is an element of Gamma) y(alpha,i) (k) f(i) (x + k) These coefficients are multivariate polynomials y(alpha,i) (x) of degree \alpha\ evaluated at lattice points k is an element of Gamma.