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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.

41A15 Spline approximation
41A25 Rate of convergence, degree of approximation