Relative Asymptotic of Multiple Orthogonal Polynomials for Nikishin Systems

TL;DR

This paper establishes the relative asymptotic behavior of ratios of multiple orthogonal polynomials associated with Nikishin systems, under specific support and perturbation conditions, advancing understanding of their asymptotic properties.

Contribution

It proves the relative asymptotic for ratios of multiple orthogonal polynomials in Nikishin systems with perturbations, extending previous results to more general measure configurations.

Findings

Established relative asymptotics for polynomial ratios

Extended asymptotic analysis to perturbed Nikishin systems

Provided conditions for measure supports and perturbations

Abstract

We prove relative asymptotic for the ratio of two sequences of multiple orthogonal polynomials with respect to Nikishin system of measures. The first Nikishin system ${\mathcal{N}}(\sigma_1,...,\sigma_m)$ is such that for each $k$, $\sigma_k$ has constant sign on its compact support $\supp {\sigma_k} \subset \mathbb{R}$ consisting of an interval $\widetilde{\Delta}_k$, on which $|\sigma_k^{\prime}| > 0$ almost everywhere, and a discrete set without accumulation points in $\mathbb{R} \setminus \widetilde{\Delta}_k$. If ${Co}(\supp {\sigma_k}) = \Delta_k$ denotes the smallest interval containing $\supp {\sigma_k}$, we assume that $\Delta_k \cap \Delta_{k+1} = \emptyset$, $k=1,...,m-1$. The second Nikishin system ${\mathcal{N}}(r_1\sigma_1,...,r_m\sigma_m)$ is a perturbation of the first by means of rational functions $r_k$, $k=1,...,m,$ whose zeros and poles lie in $\mathbb{C} \setminus \cup_{k=1}^m \Delta_k$.