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

Thomas Trogdon

Department of Applied Mathematics, University of Washington, Seattle, WA, USA

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.

Keywords: orthogonal polynomials, Riemann–Hilbert problems, steepest descent, dbar problems.

MSC: 42C05, 33C47

Received: April 5, 2025; in final form January 6, 2026; Published online January 28, 2026

Language: English

DOI: 10.3842/SIGMA.2026.006

ArXiv: 2412.18656