A note on bivariate box splines on a k-direction mesh
For \(k\geq 2\), a k-direction mesh is a set of vectors \(X=\{v^ 1,...,v^ 1,...,v^ k,...,v^ k\},\) with \(m_ 1\)-times \(v^ 1,...\), \(m_ k\)- times \(v^ k\), \(m_ i\geq 1\), where \(v^ i=(\alpha_ i,\beta_ i)\in {\mathbb{Z}}^ 2\), with \(\beta_ i\geq 0\) and \(\alpha_ i\beta_ j\neq \alpha_ j\beta_ i\) for \(i\neq j\). The author shows that the dimension of the polynomial subspace \(S_{\pi}(X)\) of the linear span of translates of bivariate box splines [see for definition \textit{C. de Boor} and \textit{R. DeVore}, Trans. Am. Math. Soc. 276, 775-788 (1983; Zbl 0529.41010)] over lattice points in \({\mathbb{Z}}^ 2\) is given by \(\dim S_{\pi}(X)=\sum_{1\leq i<j\leq k}m_ i m_ j.\)
scientific article; zbMATH DE number 3971384