Approximation by translates of refinable functions

Abstract

The functions\(f_1(x), \dots, f_r(x)\) 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\(1, x, \dots, x^{p-1}\) 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,\(r \times r\) 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\(\hat f_i(\omega)\) , but for refinable functions they can be explicitly verified from the \(c_k\).