Bivariate cubic B-splines relative to cross-cut triangulations
In this paper we prove that there are no locally supported bivariate \(C^{k-1}\) spline functions of degree k on cross-cut grid partitioned regions with no more than three lines meeting at a common vertex. We also give explicit expressions of bivariate \(C^ 1\) cubic B-spline with smallest local support on cross-cut triangular grid partitioned regions where each vertex is the intersection of three lines. Properly normalized, these B-splines are proved to be uniquely determined and form a partition of unity. Furthermore, the corresponding variation diminishing bivariate spline operators are proved to preserve all linear polynomials of two variables. These facts enable us to give error estimates for approximation by bivariate \(C^ 1\) cubic splines for functions of class C, \(C^ 1\) and \(C^ 2\).
scientific article; zbMATH DE number 3905153