Rational approximations of functions with singularities can converge at a root-exponential rate if the poles are exponentially clustered. We begin by reviewing this effect in minimax, least-squares, and AAA approximations on intervals and complex domains, conformal mapping, and the numerical solution of Laplace, Helmholtz, and biharmonic equations by the “lightning” method. Extensive and wide-ranging numerical experiments are involved. We then present further experiments giving evidence that in all of these applications, it is advantageous to use exponential clustering whose density on a logarithmic scale is not uniform but tapers off linearly to zero near the singularity. We propose a theoretical model of the tapering effect based on the Hermite contour integral and potential theory, which suggests that tapering doubles the rate of convergence. Finally we show that related mathematics applies to the relationship between exponential (not tapered) and doubly exponential (tapered) quadrature formulas. Here it is the Gauss–Takahasi–Mori contour integral that comes into play.

中文翻译：

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用于有理逼近、求积和偏微分方程的奇异点处的指数节点聚类

如果极点呈指数聚集，则具有奇点的函数的有理逼近可以以根指数速率收敛。我们首先回顾极小值、最小二乘和 AAA 近似对区间和复数域、保形映射以及拉普拉斯、亥姆霍兹和双调和方程的“闪电”方法的数值解中的这种影响。涉及广泛而广泛的数值实验。然后，我们提出进一步的实验，证明在所有这些应用中，使用指数聚类是有利的，指数聚类的对数尺度上的密度不均匀，而是在奇点附近线性逐渐减小到零。我们提出了一个基于 Hermite 轮廓积分和势理论的锥形效应的理论模型，这表明锥形可以使收敛速度加倍。最后，我们表明相关数学适用于指数（非锥形）和双指数（锥形）正交公式之间的关系。在这里，高斯-高桥-森轮廓积分开始发挥作用。