Some results on cardinal Hermite interpolation with box splines proved by the authors [Constructive Approximation 3, 223-238 (1987; Zbl 0659.41004)] in the case of a single directional derivative, are extended here for several linearly independent directions with multiplicities. Under some assumption on the smoothness of the box spline and on its defining matrix T, the cardinal Hermite interpolation problem has a system of fundamental solutions which are in \(L^{\infty}\cap L^ 2\) together with its directional derivatives. Moreover, for data sequences in \(\ell^ p(Z^ d)\), \(1\leq p\leq 2\), there is a spline function in \(L^{p'}\), \(1/p+1/p'=1\), which solves the interpolation problem.