On minimal and quasi-minimal supported bivariate splines
Let \(\Delta\) be a grid partition of the plane \({\mathbb{R}}^ 2\) with grid lines \(x=i\), \(y=i\), \(x+y=i\), and \(x-y=i\), where \(i=...,-1,0,1,..\). \(S^ k_ d(\Delta)\) denotes the space of all functions in \(C^ k({\mathbb{R}}^ 2)\) whose restriction to each triangular cell of the partition \(\Delta\) is a restriction of an element \(\pi_ d\), the space of all polynomials in x and y of total degree at most d. \(S^ k_ d(\Delta)\) is called a space of bivariate splines. The authors construct two minimal supported (ms) bivariate splines in \(S_ r=S^{3r+2}_{4r+4}(\Delta)\) and prove, surprisingly, that every ms function in this space with convex support is some constant multiple of a translate of one of them. Some further interesting results are proved; e.g. they also construct a quasiminimal supported (qms) bivariate spline in \(S_ r\), and prove that any qms function in this space with convex support is some constant multiple of a translate of this function modulo an ms function.
scientific article; zbMATH DE number 4050197
scientific article