Convergence rates of derivatives of Floater–Hormann interpolants for well-spaced nodes

Abstract

Floater–Hormann interpolants constitute a family of barycentric rational interpolants which are based on blending local polynomial interpolants of degree d. Recent results suggest that the k-th derivatives of these interpolants converge at the rate of O(hd+1−k) for k≤d as the mesh size h converges to zero. So far, this convergence rate has been proven for k=1,2 and for k≥3 under the assumption of equidistant or quasi-equidistant interpolation nodes. In this paper we extend these results and prove that Floater–Hormann interpolants and their derivatives converge at the rate of O(hjd+1−k), where hj is the local mesh size, for any k≥0 and any set of well-spaced nodes.