Convergence Rates of Derivatives of a Family of Barycentric Rational Hermite Interpolants for Well-Spaced Points

Abstract

Floater–Hormann rational interpolants and their derivatives converge as $O(h_{j}^{d+1-k})$ for any set of well-spaced points and any $0\le k\le d$ as the local mesh size $h_{j}$ converges to zero. In this paper, we extend these results to the Hermite case and investigate the approximation behavior of the k-th derivative of a recently proposed family of barycentric rational Hermite interpolants for any $k\ge 0$ and any set of well-spaced points. In addition, if the order m of the interpolated derivatives is even, the k-th derivative of the interpolants converges as $O(h_{j}^{(m+1)(d+1)-k});$ and if the order m of the interpolated derivatives is odd, the k-th derivative of the interpolants converges as $O\big (h_{j}^{(m+1)(d+1)-1-2k}\big ).$