A generalization of Floater-Hormann interpolants

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 similar to 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.. 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.

Automated summary

In this paper, the authors generalize the interpolating rational functions introduced by Floater and Hormann, resulting in a new family of rational functions depending on an additional parameter gamma. The new rational functions share many properties with the original ones when gamma > 1. They have no real poles, interpolate the given data, preserve polynomials up to a certain degree, and have a barycentric-type representation. The associated Lebesgue constants are estimated in terms of the minimum and maximum distance between consecutive nodes. It is shown that for equidistant and quasi-equidistant nodes configurations, the new interpolants have uniformly bounded Lebesgue constants for all gamma > 1.