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

Abstract

For $h>0$, degree $m\ge 0$, $0\le \beta <\pi$, and $q>0$, let ${\Sigma }_{\beta }=\{r{e}^{i\theta }:r\ge 0,|\theta |\le \beta /2\}$ 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 Möbius 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 ${\alpha }_{\beta }^{\ast }=1/[{\gamma }_{\beta }\sin (\pi /(2{\gamma }_{\beta }))]$ 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.