Convergence rates of a family of barycentric rational Hermite interpolants and their derivatives

Abstract

It is well-known that the Floater–Hormann interpolants give better results than other interpolants, especially in the case of equidistant points. In this paper, we generalize it to the Hermite case and establish a family of barycentric rational Hermite interpolants rm that do not suffer from divergence problems, unattainable points and occurrence of real poles. Furthermore, if the order m of the Hermite interpolant is even and f∈C(m+1)(d+1)+1+k[a,b], the function rm(k) converges to the corresponding function f(k) at the rate of O(h(m+1)(d+1)−k) as the mesh size h→0 for k=0,1,2, regardless of the distribution of the points; and if the interpolation points are quasi-equidistant and f∈C(m+1)(d+1)+k[a,b], the function rm(k) converges to corresponding function f(k) at the rate of O(h(m+1)(d+1)−1−2k) as h→0 for k=0,1,2, regardless of the parity of the order m of the Hermite interpolant.