Local linear independence of the translates of a box spline

Abstract

Let Ξ=(ξ _i ) ⁿ_l 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.