Convergence of repeated-pole rational approximation of the exponential on infinite sectors

Abstract

For $h>0$, degree $m\ge0$, $0\le\beta<\pi$, and $q>0$, let $\Sigma_\beta=\{re^{i\theta}:r\ge0,\ |\theta|\le\beta/2\}$ and let $E_{m,\beta}(q)$ denote the best uniform error for approximating $e^{-hz}$ on $\Sigma_\beta$ by rational functions whose only possible finite pole is $s=-q/h$, with multiplicity at most $m$. For the linear scaling $q=\alpha m$ with a fixed $\alpha>0$, a M\"obius map reduces the problem to polynomial approximation on a compact lens. Write $\gamma_\beta=2-\beta/\pi$ and let $\alpha_c(\gamma_\beta)$ denote the saddle-coalescence threshold. For $0<\alpha<\alpha_c(\gamma_\beta)$, let $H_\beta(\alpha)\in(0,1)$ 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 $E_{m,\beta}(\alpha m)\asymp m^{-1/2}H_\beta(\alpha)^m$, uniformly on compact subsets of $0<\alpha<\alpha_c(\gamma_\beta)$. On this noncoalescing interval, the exponential factor $H_\beta(\alpha)$ has the unique minimizer $\alpha_\beta^*=1/[\gamma_\beta\sin(\pi/(2\gamma_\beta))]$ and minimum $\tan(\pi/(4\gamma_\beta))$, recovering the classical half-line factor $\tan(\pi/8)=\sqrt{2}-1$ at $\beta=0$. 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.