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, 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 $h\sim h^*$). In such cases, we also estimate the uniform and the pointwise approximation errors for functions having different degree 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了推广，得到了一类依赖于额外正整数参数γ的全新有理函数族。当γ=1时，可还原为原始Floater--Hormann插值函数。针对γ>1的情形，我们证明新有理函数继承了原始Floater--Hormann函数的诸多优良性质：对于任意节点配置，它们均无极值实极点、满足给定数据的插值条件、能在特定次数以下保持多项式形式，并具有重心型表示。此外，我们根据相邻节点间的最小距离(h*)与最大距离(h)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γ>1的情况，在等距与准等距节点配置（即h~h*时）下，我们获得了一致有界的Lebesgue常数。在此类情形中，我们还对不同光滑度函数的逼近误差进行了均匀近似与逐点估计。数值实验验证了理论结果，并表明对于光滑性较差的函数，新方法的误差轮廓优于原始Floater--Hormann插值函数。