Approximation order of bivariate spline interpolation for arbitrary smoothness

By using the algorithm of Nürnberger and Riessinger (1995), we construct Hermite interpolation sets for spaces of bivariate splines Sqr(Δ1) of arbitrary smoothness defined on the uniform type triangulations. It is shown that our Hermite interpolation method yields optimal approximation order for q ⩾ 3.5r + 1. In order to prove this, we use the concept of weak interpolation and arguments of Birkhoff interpolation.