Approximation by translates of refinable functions

The functions f(1)(x),...,f(r)(x) are refinable if they are combinations 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 approximating functions with meshwidth h = 1 is a subspace of V-1 with meshwidth h = 1/2. These refinable spaces have refinable basis functions. The accuracy of the computations depends on p, the order of approximation, 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 equation-scalars in the traditional case, r x r matrices for multiwavelets. 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 Fourier transforms (f) over cap(i)(w), but for refinable functions they can be explicitly verified from the c(k).