Uniformity of Strong Asymptotics in Angelesco Systems

Abstract

Let 𝜇₁ and 𝜇₂ be two complex-valued Borel measures on the real line such that supp μ₁ = [α₁, 𝛽₁] < supp 𝜇₂ = [α₂, 𝛽₂] and d𝜇ᵢ(𝑥)=−𝜌ᵢ(𝑥)d𝑥/2πi, where 𝜌ᵢ(𝑥) is the restriction to [αᵢ, 𝛽ᵢ] of a function non-vanishing and holomorphic in some neighborhood of [αᵢ, 𝛽ᵢ]. Strong asymptotics of multiple orthogonal polynomials is considered as their multi-indices (𝑛₁, 𝑛₂) 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{𝑛₁, 𝑛₂}.