THE APPROXIMATION ORDER OF BOX SPLINE SPACES

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 Hollig (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*' and the distributional convolution phi* .