Asymptotics for Hermite-Pade rational approximants for two analytic functions with separated pairs of branch points (case of genus 0)

TL;DR

This paper analyzes the asymptotic distribution of poles in Hermite-Pade rational approximants for two functions with separated branch points, using algebraic and Riemann-Hilbert techniques to describe their behavior.

Contribution

It provides a classification of asymptotic pole distributions for Hermite-Pade approximants with separated branch points and develops strong asymptotics via a Riemann-Hilbert problem.

Findings

Pole distribution described by algebraic functions of order 3 and genus 0

Asymptotic behavior characterized by a vector-potential equilibrium problem

Strong asymptotics obtained through a 3x3 Riemann-Hilbert analysis

Abstract

We investigate the asymptotic behavior for type II Hermite-Pade approximation to two functions, where each function has two branch points and the pairs of branch points are separated. We give a classification of the cases such that the limiting counting measures for the poles of the Hermite-Pade approximants are described by an algebraic function of order 3 and genus 0. This situation gives rise to a vector-potential equilibrium problem for three measures and the poles of the common denominator are asymptotically distributed like one of these measures. We also work out the strong asymptotics for the corresponding Hermite-Pade approximants by using a 3x3 Riemann-Hilbert problem that characterizes this Hermite-Pade approximation problem.