Approximation by translates of refinable functions

Summary.

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 $c_k$.