Results on the rational approximation of functions containing singularities are presented. We build further on the ''lightning method'', recently proposed by Trefethen and collaborators, based on exponentially clustering poles close to the singularities. Our results are obtained by augmenting the lightning approximation set with either a low-degree polynomial basis or poles clustering towards infinity, in order to obtain a robust approximation of the smooth behaviour of the function. This leads to a significant increase in the achievable accuracy as well as the convergence rate of the numerical scheme. For the approximation of $x^\alpha$ on $[0,1]$, the optimal convergence rate as shown by Stahl in 1993 is now achieved simply by least-squares fitting.

翻译：本文展示了含有奇点的函数的有理逼近结果。我们进一步拓展了近期由Trefethen及其合作者提出的“闪电法”，该方法基于在奇点附近呈指数聚集的极点。通过将低次多项式基或趋向无穷的极点集群引入闪电逼近集，我们获得了对函数光滑行为的鲁棒逼近。这一改进显著提升了数值格式的收敛速度与可达精度。对于区间$[0,1]$上的函数$x^\alpha$逼近，如今仅通过最小二乘拟合即可实现Stahl于1993年证明的最优收敛速率。