Local linear independence of the translates of a box spline

Let Ξ=(ξ _i ) _l ⁿ be a sequence of vectors inR ^m . The box splineM _Ξ is defined as the distribution given by $$M_\Xi :\varphi \to \int_{0,1]^n } \varphi \left( {\sum\limits_{i = 1}^n {\lambda (i)\xi _i } } \right)d\lambda ,\varphi \in C_c^\infty (R^m ).$$ . Suppose that Ξ contains a basis forR ^m . ThenM _Ξ∈L _∞(R ^m ). Assume $$\Xi \subset V: = z^m .$$ . Consider the translatesM _v :=M _Ξ(·?v),v∈V. It is known that (M _v ) _V is linearly dependent unless (*) $$|\det Z| = 1forallbasesZ \subset \Xi$$ . This paper demonstrates that under condition (*), (M _v ) _V is locally linearly independent, i.e., $$\{ M_v ;\sup p M_v \cap A \ne \not 0\}$$ is linearly independent over any open setA.