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

arXiv:2610.09251v1 Announce Type: new Abstract: For $h>0$, degree $m\ge0$, $0\le\beta0$, 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