Box Splines with Rational Directions and Linear Diophantine Equations

A necessary and sufficient condition for the linear independence of integer translates of Box splines with rational directions is presented in terms of intrinsic properties of the defining matrices. We also give a necessary and sufficient condition for the space of linear dependence relations to be finite dimensional. A method to compute the approximation order of these Box spline spaces is obtained. All these conditions can be tested by finite steps of computations based on elementary properties of the matrices. The method of proofs is from linear diophantine equations.