A note on generalized Floater–Hormann interpolation at arbitrary distributions of nodes

The paper is concerned with a generalization of Floater–Hormann (briefly FH) rational interpolation recently introduced by the authors. Compared with the original FH interpolants, the generalized ones depend on an additional integer parameter \(\gamma >1\) $\gamma >1$ , that, in the limit case \(\gamma =1\) $\gamma =1$ returns the classical FH definition. Here we focus on the general case of an arbitrary distribution of nodes and, for any \(\gamma >1\) $\gamma >1$ , we estimate the sup norm of the error in terms of the maximum (h) and minimum ( \(h^*\) $h^{\ast }$ ) distance between two consecutive nodes. In the special case of equidistant ( \(h=h^*\) $h=h^{\ast }$ ) or quasi–equidistant ( \(h\approx h^*\) $h\approx h^{\ast }$ ) nodes, the new estimate improves previous results requiring some theoretical restrictions on \(\gamma \) $\gamma$ which are not needed as shown by the numerical tests carried out to validate the theory.