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

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