Uniformity of Strong Asymptotics in Angelesco Systems

TL;DR

This paper investigates the uniformity of strong asymptotics for multiple orthogonal polynomials associated with Angelesco systems, ensuring error terms remain controlled as the polynomial degrees grow large in both indices.

Contribution

It establishes that the error terms in the asymptotic formulas are uniform with respect to the minimum of the multi-indices in Angelesco systems.

Findings

Error terms are uniform as degrees tend to infinity.

Strong asymptotics hold uniformly across multi-indices.

Results apply to measures with non-vanishing holomorphic densities.

Abstract

Let $\mu_1$ and $\mu_2$ be two complex-valued Borel measures on the real line such that $\operatorname{supp} \mu_1 =[\alpha_1,\beta_1] < \operatorname{supp} \mu_2 =[\alpha_2,\beta_2]$ and ${\rm d}\mu_i(x) = -\rho_i(x){\rm d}x/2\pi {\rm i}$, where $\rho_i(x)$ is the restriction to $[\alpha_i,\beta_i]$ of a function non-vanishing and holomorphic in some neighborhood of $[\alpha_i,\beta_i]$. Strong asymptotics of multiple orthogonal polynomials is considered as their multi-indices $(n_1,n_2)$ tend to infinity in both coordinates. The main goal of this work is to show that the error terms in the asymptotic formulae are uniform with respect to $\min\{n_1,n_2\}$.