Some results on cubic and quartic quasi-interpolation of optimal approximation order on type-1 triangulations

t. In this paper we summarize the results on new families of C1 quartic and cubic 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 reproduce cubic and quadratic polynomials and yield approximation order four and three for smooth functions, respectively.