APPROXIMATION BY TRANSLATES OF REFINABLE FUNCTIONS

The functions f(1)(x),...,f(r)(x) are refinable if they are combinatio ns of the rescaled and translated functions f(i)(2x - k). This is very common in scientific computing on a regular mesh. The space V-0 of ap proximating functions with meshwidth h = 1 is a subspace of V-1 with m eshwidth h = 1/2. These refinable spaces have refinable basis function s. The accuracy of the computations depends on p, the order of approxi mation, which is determined by the degree of polynomials 1,x,...,x(p-1 ) that lie in V-0. Most refinable functions (such as scaling functions in the theory of wavelets) have no simple formulas. The functions f(i )(x) are known only through the coefficients c(k) in the refinement eq uation-scalars in the traditional case, r x r matrices for multiwavele ts. The scalar ''sum rules'' that determine p are well known. We find the conditions on the matrices c(k) that yield approximation of order p from V-0 These are equivalent to the Strang-Fix conditions on the Fo urier transforms (f) over cap(i)(w), but for refinable functions they can be explicitly verified from the c(k).