Quadratic Hermite-Padé approximation to the exponential function: A Riemann-Hilbert approach

摘要

We investigate the asymptotic behavior of the polynomials p, q, r of degrees n in type I Hermite-Pade approximation to the exponential function, defined by p(z)e^{-z} + q(z) + r(z)e^z = O(z^{3n+2}) as z --> 0. These polynomials are characterized by a Riemann-Hilbert problem for a 3 x 3 matrix valued function. We use the Deift-Zhou steepest descent method for Riemann-Hilbert problems to obtain strong uniform asymptotics for the scaled polynomials p(3nz), q(3nz), and r(3nz) in every domain in the complex plane. An important role is played by a three-sheeted Riemann surface and certain measures and functions derived from it. Our work complements the recent results of Herbert Stahl.