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

摘要

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 r(m) 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 is an element of is an element of C(m+1)(d+1)+1+k[a, b], the function r(m)((k)) m 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 is an element of C(m+1)(d+1)+k[a, b], the function r(m)((k)) m 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. (C) 2021 Elsevier B.V. All rights reserved.