Enhancing the approximation order of local Shepard operators by Hermite polynomials

We show how to combine local Shepard operators with Hermite polynomials on the simplex [C. K. Chui, M.-J. Lai, Multivariate vertex splines and finite elements, J. Approx. Theory 60 (1990) 245–343] so as to raise the algebraic precision of the Shepard–Taylor operators [R. Farwig, Rate of convergence of Shepard’s global interpolation formula, Math. Comp. 46 (1986) 577–590] that use the same data and contemporaneously maintain the interpolation properties at each sample point (derivative data included) and a good accuracy of approximation. Numerical results are provided.