Riemann–Hilbert analysis and uniform convergence of rational interpolants to the exponential function

Abstract

We study the asymptotic behavior of the polynomials p and q of degrees n, rational interpolants to the exponential function, defined by p ( z ) e - z / 2 + q ( z ) e z / 2 = O ( ω 2 n + 1 ( z ) ) , as z tends to the roots of ω 2 n + 1 , a complex polynomial of degree 2 n + 1 . The roots of ω 2 n + 1 may grow to infinity with n, but their modulus should remain uniformly bounded by c log ( n ) , c < 1 / 2 , as n → ∞ . We follow an approach similar to the one in a recent work with Arno Kuijlaars and Walter Van Assche on Hermite–Padé approximants to e z . The polynomials p and q are characterized by a Riemann–Hilbert problem for a 2 × 2 matrix valued function. The Deift–Zhou steepest descent method for Riemann–Hilbert problems is used to obtain strong uniform asymptotics for the scaled polynomials p ( 2 nz ) and q ( 2 nz ) in every domain in the complex plane. From these asymptotics, we deduce uniform convergence of general rational interpolants to the exponential function and a precise estimate on the error function. This extends previous results on rational interpolants to the exponential function known so far for real interpolation points and some cases of complex conjugate interpolation points.