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.

翻译：本文对Floater与Hormann提出的有理插值函数进行推广，引入了一个新的有理函数族，该族依赖于附加正整数参数$\gamma$。当$\gamma=1$时，可还原为原始Floater-Hormann插值函数。我们证明了对于$\gamma>1$的情况，新有理函数保留了原始Floater-Hormann函数的诸多优良性质：在紧致区间上任意节点配置下均无实极点、可插值给定数据、保持至特定次数的多项式，并具有重心型表示形式。进一步地，我们以相邻节点间最小距离($h^*$)与最大距离($h$)为参数，估计了相应的Lebesgue常数。结果表明，与原始Floater-Hormann插值函数不同，对于所有$\gamma>1$，在等距或准等距节点配置（即$h\sim h^*$）下，Lebesgue常数一致有界。对于此类节点配置，我们证明了当节点数趋于无穷时，对于任意连续函数$f$及所有$\gamma>1$，新插值函数（$\gamma>1$）一致收敛于被插值函数$f$，而原始FH插值函数（$\gamma=1$）无法保证该性质。此外，我们给出了不同光滑度函数的逼近误差的一致估计与逐点估计。数值实验验证了理论结果，并表明对于光滑性较差的函数，新方法相较原始Floater-Hormann插值函数具有更优的误差分布。