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

Abstract

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 \(\gamma >1\), that, in the limit case \(\gamma =1\) returns the classical FH definition. Here we focus on the general case of an arbitrary distribution of nodes and, for any \(\gamma >1\), we estimate the sup norm of the error in terms of the maximum (h) and minimum (\(h^*\)) distance between two consecutive nodes. In the special case of equidistant (\(h=h^*\)) or quasi–equidistant (\(h\approx h^*\)) nodes, the new estimate improves previous results requiring some theoretical restrictions on \(\gamma \) which are not needed as shown by the numerical tests carried out to validate the theory.