On the approximation power of bivariate splines

We show how to construct stable quasi-interpolation schemes in the bivariate spline spaces Sdr(Δ) with d⩾ 3r + 2 which achieve optimal approximation order. In addition to treating the usual max norm, we also give results in the Lp 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.