Uniformity of Strong Asymptotics in Angelesco Systems

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}.