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arXiv · 2412.18656

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

Thomas Trogdon
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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.

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Thomas Trogdon. 2024-12-24. On the Asymptotics of Orthogonal Polynomials on Multiple Intervals with Non-Analytic Weights. https://doi.org/10.3842/sigma.2026.006

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