On the approximation power of bivariate splines

We show how to construct stable quasi-interpolation schemes in the bivariate spline spaces S _d ^r (Δ) with d⩾ 3r + 2 which achieve optimal approximation order. In addition to treating the usual max norm, we also give results in the L _p norms, and show that the methods also approximate derivatives to optimal order. We pay special attention to the approximation constants, and show that they depend only on the smallest angle in the underlying triangulation and the nature of the boundary of the domain. This revised version was published online in August 2006 with corrections to the Cover Date.