Optimal quasi-interpolation by quadratic C1 splines on four-directional meshes. Approximation Theory XI: Gatlinburg 2004

Abstract. We describe a new scheme based on quadratic C1-splines on type-2 triangulations approximating gridded data. The quasi-interpolating splines are directly determined by setting the Bernstein-Bézier coefficients of the splines to appropriate combinations of the given data values. In this way, each polynomial piece of the approxi-mating 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. The aim of this paper is to describe local methods which use qua-dratic C1-splines on the type-2 triangulation to approximate data on