Local linear independence of the translates of a box spline
Let \(\Xi =(\xi_ i)^ n_ 1\) be a sequence of vectors in \(R^ m\). The box spline \(M_{\Xi}\) is defined as the distribution given by \[ M_{\Xi}: \phi \to \int_{[0,1]^ n}\phi (\sum^{n}_{i=1}\lambda (i)\xi_ i)d\lambda,\quad \phi \in C_ c^{\infty}(R^ m). \] Suppose that \(\Xi\) contains a basis for \(R^ m\). Then \(M_{\Xi}\in L_{\infty}(R^ m)\). Assume \(\Xi \subset V:=Z^ m\). Consider the translates \(Mv:=M_{\Xi}(\cdot -v)\), \(v\in V\). It is known that \((Mv)_ V\) is linearly dependent unless (*) \(| \det Z| =1\) for all bases \(Z\subset \Xi\). This paper demonstrates that under condition (*), \((Mv)_ V\) is locally linearly independent, i.e., \(\{\) Mv;supp Mv\(\cap A\neq \emptyset \}\) is linearly independent over any open set A.
scientific article; zbMATH DE number 3939905