Quasi-interpolation by C1quartic splines on type-1 triangulations

In this paper we construct two new families of quartic quasi-interpolating splines on type-1 triangulations approximating regularly distributed data. The splines are directly determined by setting their Bernstein–Bézier coefficients to appropriate combinations of the given data values instead of defining the approximating splines as linear combinations of compactly supported bivariate spanning functions and do not use prescribed derivatives at any point of the domain. The quasi-interpolation operators provided by the proposed schemes interpolate the data values at the vertices of the triangulation, reproduce cubic polynomials and yield approximation order four for smooth functions. We also propose some numerical tests that confirm the theoretical results. The research that led to the present paper was partially supported by a grant of the group GNCS of INdAM.