Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 22 (2026), 006, 42 pages      arXiv:2412.18656      https://doi.org/10.3842/SIGMA.2026.006

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

Received April 05, 2025, in final form January 06, 2026; Published online January 28, 2026

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.

Key words: orthogonal polynomials; Riemann-Hilbert problems; steepest descent; dbar problems.

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