Accuracy of lattice translates of several multidimensional refinable functions

Complex-valued functions f(1),..., f(r) on R(d) are refinable if they are linear combinations of finitely many of the rescaled and translated functions f(i)(Ax - k), where the translates k are taken along a lattice Gamma subset of R(d) and A is a dilation matrix that expansively maps Gamma into itself. Refinable functions satisfy a refinement equation 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 such that all multivariate polynomials q with degree(q) < p are exactly reproduced from linear combinations of translates of f(1,) ..., f(r) along 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 element of Gamma)gamma(alpha,i)(k)f(i)(x + k). These coefficients are multivariate polynomials gamma(alpha,i)(x) Of degree \alpha\ evaluated at lattice points k is an element of Gamma. (C) 1998 Academic Press.