Relative Asymptotic of Multiple Orthogonal Polynomials for Nikishin Systems

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 $\supp {σ_k} \subset \mathbb{R}$ consisting of an interval ${\tilde{\Delta }}_k$, on which $|{\sigma }_k^{\prime }|>0$ almost everywhere, and a discrete set without accumulation points in $\mathbb{R}\setminus {\tilde{\Delta }}_k$. If ${Co}(\supp {σ_k}) = Δ_k$ denotes the smallest interval containing $\supp {σ_k}$, we assume that ${\Delta }_k\cap {\Delta }_{k+1}=\varnothing$, $k=1,...,m-1$. 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 $\mathbb{C}\setminus {\cup }_{k=1}^m{\Delta }_k$.

30 pages