Optimal Quasi-Interpolation by Quadratic C-Splines on Type-2 Triangulations

We describe a new scheme based on quadratic C-splines on type-2 triangulations approximating gridded data. The quasiinterpolating splines are directly determined by setting the BernsteinBézier coefficients of the splines to appropriate combinations of the given data values. In this way, each polynomial piece of the approximating spline is immediately available from local portions of the data, without using prescribed derivatives at any point of the domain. Since the Bernstein-Bézier coefficients of the splines are computed directly, an intermediate step making use of certain locally supported splines spanning the space is not needed. We prove that the splines yield optimal approximation order for smooth functions, where we provide explicit constants in the corresponding error bounds.