Let M M be a box spline associated with an arbitrary set of directions and suppose that S ( M ) S(M) is the space spanned by the integer translates of M M . In this note, the subspace of all polynomials in S ( M ) 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 ) S(M) to smooth functions is thereby characterized. This extends a well-known result of de Boor and Höllig ( B 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 ϕ \phi , between the semidiscrete convolution ϕ ∗ ′ \phi \ast ’ and the distributional convolution ϕ ∗ \phi \ast .