# zbMATH — the first resource for mathematics

On a bivariate B-spline basis. (English) Zbl 0559.41010
While there does not exist any nontrivial locally supported bivariate $$C^{k-1}$$ spline function of degree k on a triangulation refinement $$\Delta_{mn}$$ of a rectangular grid partition for $$k\geq 3$$, a basis of bivariate $$C^ 1$$ quadratic B-splines with smallest symmetric supports is given when the grid partition is uniform. In addition, B-spline identities which include a generalization of Marsden’s identity for univariate splines are given. These identities enable us to give error estimates for approximation from the entire space of $$C^ 1$$ quadratic spline functions with grid partition $$\Delta_{mn}$$ and to give asymptotic formulas.

##### MSC:
 41A15 Spline approximation 41A25 Rate of convergence, degree of approximation