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Relative asymptotics of multiple orthogonal polynomials for Nikishin systems of two measures

Abey López-García, Guillermo López Lagomasino

TL;DR

The paper addresses relative asymptotics for two Nikishin systems of two measures, where the second system perturbs the first by nonnegative integrable weights. It develops a fixed-point Szegő-function framework: a contraction on a space of vector Szegő functions $\Phi=(\Phi_1,\Phi_2)$ yields the limiting behavior of the perturbed multiple orthogonal polynomials relative to the unperturbed ones, namely $\lim_{\mathbf n\in\Lambda} \frac{\widetilde{Q}_{{\bf n},k}(z)}{Q_{{\bf n},k}(z)}=\frac{\Phi_k(z)}{\Phi_k(\infty)}$ for $z$ outside the corresponding supports, with analogous limits for the associated second-kind functions. The results are expressed via the fixed-point relations $\Phi=T(\Phi)$ and detailed Szegő-function constructions $\mathsf{G}_{\Delta_k}$, building upon and extending the scalar relative asymptotics literature (Szegő, MNT, Gui, Berndtsson–Lopéz) to a two-measure Nikishin setting. The work also outlines the potential extension to $m\ge 3$ measures, noting a technical obstruction in uniqueness of fixed points for the higher-dimensional contraction. Overall, the findings provide explicit Szegő-type limits that quantify how nonnegative perturbations of Nikishin-system measures control the asymptotics of the corresponding system of multiple orthogonal polynomials.

Abstract

We study the relative asymptotics of two sequences of multiple orthogonal polynomials corresponding to two Nikishin systems of measures on the real line, the second one of which is obtained from the first one perturbing the generating measures with non-negative integrable functions. Each Nikishin system consists of two measures.

Relative asymptotics of multiple orthogonal polynomials for Nikishin systems of two measures

TL;DR

The paper addresses relative asymptotics for two Nikishin systems of two measures, where the second system perturbs the first by nonnegative integrable weights. It develops a fixed-point Szegő-function framework: a contraction on a space of vector Szegő functions yields the limiting behavior of the perturbed multiple orthogonal polynomials relative to the unperturbed ones, namely for outside the corresponding supports, with analogous limits for the associated second-kind functions. The results are expressed via the fixed-point relations and detailed Szegő-function constructions , building upon and extending the scalar relative asymptotics literature (Szegő, MNT, Gui, Berndtsson–Lopéz) to a two-measure Nikishin setting. The work also outlines the potential extension to measures, noting a technical obstruction in uniqueness of fixed points for the higher-dimensional contraction. Overall, the findings provide explicit Szegő-type limits that quantify how nonnegative perturbations of Nikishin-system measures control the asymptotics of the corresponding system of multiple orthogonal polynomials.

Abstract

We study the relative asymptotics of two sequences of multiple orthogonal polynomials corresponding to two Nikishin systems of measures on the real line, the second one of which is obtained from the first one perturbing the generating measures with non-negative integrable functions. Each Nikishin system consists of two measures.
Paper Structure (5 sections, 8 theorems, 131 equations)

This paper contains 5 sections, 8 theorems, 131 equations.

Table of Contents

  1. Introduction
  2. Background
  3. Nikishin systems of two measures and multiple orthogonal polynomials
  4. Auxiliary results
  5. Proof of Theorem \ref{['theomain']}

Key Result

Theorem 1.3

Suppose that $\sigma_{k}'>0$ almost everywhere on $\Delta_{k}$, $k=1,2$. Let $\rho_{k}$ be non-negative measurable functions on $\Delta_{k}$ such that $\rho_{k}\in L^{1}(\sigma_{k})$, and assume that there exists a polynomial $q\not\equiv 0$ such that $|q|\rho_k^{\pm 1} \in L^{\infty}(\sigma_k)$ for uniformly on compact subsets of the indicated regions.

Theorems & Definitions (20)

  • Definition 1.1
  • Definition 1.2
  • Theorem 1.3
  • Theorem 2.1
  • proof
  • Definition 2.2
  • Lemma 2.3
  • Lemma 2.4
  • Definition 2.5
  • Theorem 2.6
  • ...and 10 more