Uniform Chebyshev asymptotics for repeated-pole rational approximation of the exponential

Abstract

We study uniform approximation of $\exp (tz)$, $t>0$, on $(-\infty ,0]$ by rational functions $P_m(z)/(q_m-z{)}^m$ with a prescribed repeated pole $q_m>0$. A Möbius transformation reduces the problem to polynomial approximation of $F_{\lambda }(x)=\exp \left(-\lambda \frac{1-x}{1+x}\right)$ on $[-1,1]$, where $\lambda =t{q}_m$. We derive a uniform two-saddle asymptotic formula for the Chebyshev coefficients, including explicit amplitude and phase and a relative remainder for each localized complex saddle contribution, covering fixed, sublinear, and linear pole scalings away from saddle coalescence. Because the two saddle contributions can cancel in a single coefficient, we pass to a growing block of neighboring coefficients and prove that the whole block cannot cancel. This transfers the coefficient asymptotics to approximation errors. For $q_m=(\alpha /t)m$, $0<\alpha <3\sqrt{3}/2$, the best uniform error has two-sided order $m^{-1/2}{H}_e(\alpha {)}^m$; at the optimal ratio $\alpha =1/\sqrt{2}$ this becomes $m^{-1/2}(\sqrt{2}-1{)}^m$. The normalized Chebyshev-weighted $L^2$ projection error has an explicit bounded oscillatory profile. These prefactor-resolved estimates yield a two-term precision-to-work law and quantify mismatch between pole-design and stopping degrees. Finally, the scalar error gives an exact worst-case matrix-action benchmark for self-adjoint negative semidefinite matrices and dimension-independent shift-and-invert Krylov bounds.