Convergence rates of a Hermite generalization of Floater-Hormann interpolants

Cirillo and Hormann (2018) introduce an iterative approach to the Hermite interpolation problem, which, starting from the Lagrange interpolant, successively adds m corrections terms to interpolate the data up to the mth derivative. The method is general enough to be applied to any interpolant in linear form with a sufficiently continuous set of basis functions, but Cirillo and Hormann focus their attention on Floater-Hormann interpolants, a family of barycentric rational interpolants that are based on a particular blend of local polynomial interpolants of degree d. They show that the resulting iterative rational Hermite interpolants converge at the rate of O(h((m+1)(d+1))) as the mesh size h converges to zero for m = 1, 2, and their numerical results suggest that the same rate holds for m > 2. In this paper we prove this convergence rate for any m >= 1. (C) 2019 Elsevier B.V. All rights reserved.