Strong asymptotics of multi-level Hermite-Pad\'e polynomials

TL;DR

This paper derives strong asymptotic formulas for multiple orthogonal polynomials in Hermite-Padé approximation, enabling precise convergence estimates and insights into Cauchy biorthogonal polynomials within Nikishin systems.

Contribution

It provides the first comprehensive strong asymptotics for multi-level Hermite-Padé polynomials in Nikishin systems, advancing understanding of their convergence and behavior.

Findings

Exact convergence rate estimates for rational approximations

Strong asymptotics for Cauchy biorthogonal polynomials

Enhanced understanding of multi-level Hermite-Padé polynomials

Abstract

We obtain the strong asymptotics of multiple orthogonal polynomials which arise in a mixed type Hermite-Pad\'e approximation problem defined on a Nikishin system of functions. The results obtained allow to give exact estimates of the rate of convergence of the approximating rational functions and the strong asymptotics of Cauchy biorthogonal polynomials.