A generalization of Floater--Hormann interpolants

Abstract

In this paper the interpolating rational functions introduced by Floater and Hormann are generalized leading to a whole new family of rational functions depending on $\gamma$, an additional positive integer parameter. For $\gamma = 1$, the original Floater--Hormann interpolants are obtained. When $\gamma>1$ we prove that the new rational functions share a lot of the nice properties of the original Floater--Hormann functions. Indeed, for any configuration of nodes in a compact interval, they have no real poles, interpolate the given data, preserve the polynomials up to a certain fixed degree, and have a barycentric-type representation. Moreover, we estimate the associated Lebesgue constants in terms of the minimum ($h^*$) and maximum ($h$) distance between two consecutive nodes. It turns out that, in contrast to the original Floater-Hormann interpolants, for all $\gamma > 1$ we get uniformly bounded Lebesgue constants in the case of equidistant and quasi-equidistant nodes configurations (i.e., when $h\sim h^*$). For such configurations, as the number of nodes tends to infinity, we prove that the new interpolants ($\gamma>1$) uniformly converge to the interpolated function $f$, for any continuous function $f$ and all $\gamma>1$. The same is not ensured by the original FH interpolants ($\gamma=1$). Moreover, we provide uniform and pointwise estimates of the approximation error for functions having different degrees of smoothness. Numerical experiments illustrate the theoretical results and show a better error profile for less smooth functions compared to the original Floater-Hormann interpolants.