发表机构
Clemson University; Zhejiang University of Finance and Economics（克莱姆森大学; 浙江财经大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文研究无限扇区上指数函数重极点有理逼近的收敛性，通过Möbius映射和双鞍点展开得到尖锐的指数收敛阶，并给出最优缩放参数，统一了经典半直线结果。
AI 中文摘要
对于$h>0$，次数$m\ge0$，$0\le\beta<\pi$和$q>0$，设$\Sigma_\beta=\{re^{i\theta}:r\ge0,\\ |\theta|\le\beta/2\}$，并设$E_{m,\beta}(q)$表示在$\Sigma_\beta$上用仅可能有限极点为$s=-q/h$（重数至多为$m$）的有理函数逼近$e^{-hz}$的最佳一致误差。对于固定$\alpha>0$的线性缩放$q=\alpha m$，一个Möbius映射将问题归结为紧致透镜上的多项式逼近。记$\gamma_\beta=2-\beta/\pi$，并设$\alpha_c(\gamma_\beta)$表示鞍点合并阈值。对于$0<\alpha<\alpha_c(\gamma_\beta)$，设$H_\beta(\alpha)\in(0,1)$为实鞍点作用的指数。一个精确的边界Faber积分、一个全局分数幂围道变形和一个双鞍点展开共同给出了尖锐的双边阶$E_{m,\beta}(\alpha m)\asymp m^{-1/2}H_\beta(\alpha)^m$，该结果在$0<\alpha<\alpha_c(\gamma_\beta)$的紧致子集上一致成立。在此非合并区间上，指数因子$H_\beta(\alpha)$具有唯一极小点$\alpha_\beta^*=1/[\gamma_\beta\sin(\pi/(2\gamma_\beta))]$和最小值$\tan(\pi/(4\gamma_\beta))$，在$\beta=0$时恢复了经典的半直线因子$\tan(\pi/8)=\sqrt{2}-1$。数值范围估计给出了投影移位-反转Arnoldi方法的矩阵维度无关上界，而有限次极点搜索和非正规实验则说明了扇区一致律与单个矩阵计算之间的区别。
英文摘要
For $h>0$, degree $m\ge0$, $0\leβ<π$, and $q>0$, let $Σ_β=\{re^{iθ}:r\ge0,\ |θ|\leβ/2\}$ and let $E_{m,β}(q)$ denote the best uniform error for approximating $e^{-hz}$ on $Σ_β$ by rational functions whose only possible finite pole is $s=-q/h$, with multiplicity at most $m$. For the linear scaling $q=αm$ with a fixed $α>0$, a Möbius map reduces the problem to polynomial approximation on a compact lens. Write $γ_β=2-β/π$ and let $α_c(γ_β)$ denote the saddle-coalescence threshold. For $0<α<α_c(γ_β)$, let $H_β(α)\in(0,1)$ be the exponential of the real saddle action. An exact boundary Faber integral, a global fractional-power contour deformation, and a two-saddle expansion yield the sharp two-sided order $E_{m,β}(αm)\asymp m^{-1/2}H_β(α)^m$, uniformly on compact subsets of $0<α<α_c(γ_β)$. On this noncoalescing interval, the exponential factor $H_β(α)$ has the unique minimizer $α_β^*=1/[γ_β\sin(π/(2γ_β))]$ and minimum $\tan(π/(4γ_β))$, recovering the classical half-line factor $\tan(π/8)=\sqrt{2}-1$ at $β=0$. Numerical-range estimates give matrix dimension-independent upper bounds for projected shift-and-invert Arnoldi, while finite-degree pole searches and nonnormal experiments illustrate the distinction between the sector-uniform law and individual matrix computations.
