Abstract
Let A be a dilation matrix, an n x n expansive matrix that maps Z(n) into itself. Let. be a finite subset of Z(n), and for kappa is an element of Lambda let c kappa be r x r complex matrices. The refinement equation corresponding to A, Z(n),Lambda and c = {c(kappa)}(kappa is an element of Lambda) is f(x) = Sigma(kappa is an element of Lambda)c(kappa) f(Ax - kappa). A solution f : R-n -> C-r, if one exists, is called a refinable vector function or a vector scaling function of multiplicity r. This paper characterizes the higher-order smoothness of compactly supported solutions of the refinement equation, in terms of the p-norm joint spectral radius of a finite set of finite matrices determined by the coefficients c(kappa).