The approximation order of box spline spaces

Abstract:

Let $M$ be a box spline associated with an arbitrary set of directions and suppose that $S(M)$ is the space spanned by the integer translates of $M$. In this note, the subspace of all polynomials in $S(M)$ is shown to be the joint kernal of a certain collection of homogeneous differential operators with constant coefficients. The approximation order from the dilates of $S(M)$ to smooth functions is thereby characterized. This extends a well-known result of de Boor and Höllig ($B$-splines from parallelepipeds, J. Analyse Math. 42 (1982/83), 99-115), on box splines with integral direction sets. The argument used is based on a new relation, valid for any compactly supported distribution $\phi$, between the semidiscrete convolution $\phi \ast ’$ and the distributional convolution $\phi \ast$.