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：本文提出了包含奇点函数的合理近似结果。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。通过使用低次多项式基或向无穷远处聚集的极点来增强闪电近似集，我们获得了对函数光滑行为的鲁棒逼近。这使得数值方案的收敛速度与可实现的精度显著提升。对于$[0,1]$区间上$x^\alpha$的逼近，1993年Stahl所证明的最优收敛速率现已通过简单的最小二乘拟合即可实现。