Some characterizations for box spline wavelets and linear diophantine equations

Box splines are investigated from the point of view of wavelets. Some characterizations concerning linear independence of integer translates of Box splines are presented in terms of the defining matrices. It is shown that a direct extension of a criterion for linear independence of refinable functions in the univariate case to the multivariate case holds for the Box spline M_Ξ in R^s when rankΞ = s while not any more when rank Ξ <s.