Riemann-Hilbert analysis and uniform convergence of rational interpolants to the exponential function
The polynomials \(p\) and \(q\) of degrees \(n\) constituting rational approximants to the exponential function satisfy the relation \(p(z)\exp(-z/2)+q(z)\exp(z/2)= O(\omega_{2n+1}(z))\), as \(z\) tends to the roots of the complex polynomial \(\omega_{2n+1}(z)\). The large \(n\) asymptotics is studied of the polynomials \(p\) and \(q\), which are characterized by a Riemann-Hilbert problem for a \(2\times2\) matrix valued function. The Deift-Zhou steepest descent method for Riemann-Hilbert problems is used to obtain strong uniform asymptotics for the scaled polynomials \(p(2nz)\) and \(q(2nz)\) in every domain of the complex plane. A precise estimate of the error function is given.
scientific article; zbMATH DE number 2136998
scientific article