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

Abstract

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