On the Asymptotics of Orthogonal Polynomials on Multiple Intervals with Non-Analytic Weights

TL;DR

This paper studies the asymptotic behavior of orthogonal polynomials on multiple intervals with non-analytic weights, extending previous results to less regular perturbations using advanced Riemann-Hilbert analysis.

Contribution

It extends asymptotic analysis of orthogonal polynomials to measures with only differentiability, not analyticity, using the Deift-Zhou method and its extensions.

Findings

Extended asymptotic results to non-analytic weights with bounded second derivatives.

Provided error bounds similar to Chebyshev-like measures.

First known asymptotic analysis for measures with only differentiability.

Abstract

We consider the asymptotics of orthogonal polynomials for measures that are differentiable, but not necessarily analytic, multiplicative perturbations of Jacobi-like measures supported on disjoint intervals. We analyze the Fokas-Its-Kitaev Riemann-Hilbert problem using the Deift-Zhou method of nonlinear steepest descent and its $\overline{\partial}$ extension due to Miller and McLaughlin. Our results extend that of Yattselev in the case of Chebyshev-like measures with error bounds that give similar rates while allowing less regular perturbations. For the general Jacobi-like case, we present, what appears to be the first result for asymptotics when the perturbation of the measure is only assumed to be differentiable with bounded second derivative.