2023/07/11 by Themistoclakis, Woula, Van Barel, Marc
#41A05 (Primary) 41A20 #41A25 (Secondary) #FOS: Mathematics #Numerical Analysis (math.NA)
paper · doi:10.48550/arxiv.2307.05345
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 γ, an additional positive integer parameter. For γ= 1, the original Floater--Hormann interpolants are obtained. When γ>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 γ> 1 we get uniformly bounded Lebesgue constants in the case of equidistant and quasi-equidistant nodes configurations (i.e., when h∼ h^*). For such configurations, as the number of nodes tends to infinity, we prove that the new interpolants (γ>1) uniformly converge to the interpolated function f, for any continuous function f and all γ>1. The same is not ensured by the original FH interpolants (γ=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.