Ratio Asymptotic of Hermite-Pad\'e Orthogonal Polynomials for Nikishin Systems. II

TL;DR

This paper establishes the ratio asymptotic behavior of Hermite-Padé orthogonal polynomials associated with Nikishin systems, extending understanding of their asymptotic properties in complex approximation theory.

Contribution

It proves ratio asymptotics for multiple orthogonal polynomials related to Nikishin systems with specific support conditions, advancing theoretical knowledge in orthogonal polynomial asymptotics.

Findings

Proved ratio asymptotic for Hermite-Padé polynomials in Nikishin systems.

Extended asymptotic analysis to measures with support on intervals and isolated points.

Enhanced understanding of polynomial behavior in complex approximation contexts.

Abstract

We prove ratio asymptotic for sequences of multiple orthogonal polynomials with respect to a Nikishin system of measures ${\mathcal{N}}(\sigma_1,...,\sigma_m)$ such that for each $k$, the support of $\sigma_k$ consists of an interval $\widetilde{\Delta}_k$, on which $\sigma_k^{\prime} > 0$ almost everywhere, and a set without accumulation points in $\mathbb{R} \setminus \widetilde{\Delta}_k$.