A system of differential equations connected with a polynomial class of shifts of a box spline (Q923224)

We introduce a system of linear partial differential equations, the solutions of the corresponding homogeneous system of which are polynomials, decomposable into shifts of a box spline. A uniqueness and existence theorem is proved for such systems. A method is presented of selecting independent initial conditions. It is of interest that these initial conditions, in their turn, prove to be tightly connected with the basis space of the spline, composed of B-splines. We will also give a characterization of the space of polynomials, isomorphic to this space of splines. As corollaries, some properties are obtained of the polynomial class of the box spline, and the dimensions of the spaces of splines are computed as well in some special cases.