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On smooth perturbations of Chebyshëv polynomials and $ \bar {\partial } $-Riemann–Hilbert method

Part of: Nontrigonometric harmonic analysis

Published online by Cambridge University Press:  24 February 2022

Maxim L. Yattselev
Maxim L. Yattselev*
Affiliation:
Department of Mathematical Sciences, Indiana University–Purdue University Indianapolis, 402 North Blackford Street, Indianapolis, IN 46202, USA
*
e-mail: [email protected]
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Abstract

$\bar {\partial } $ -extension of the matrix Riemann–Hilbert method is used to study asymptotics of the polynomials $ P_n(z) $ satisfying orthogonality relations

$$ \begin{align*} \int_{-1}^1 x^lP_n(x)\frac{\rho(x)dx}{\sqrt{1-x^2}}=0, \quad l\in\{0,\ldots,n-1\}, \end{align*} $$

where $ \rho (x) $ is a positive $ m $ times continuously differentiable function on $ [-1,1] $ , $ m\geq 3 $ .

Keywords

Orthogonal polynomialsstrong asymptoticsmatrix Riemann–Hilbert approach

MSC classification

Primary: 42C05: Orthogonal functions and polynomials, general theory
Type
Article
Information
Canadian Mathematical Bulletin , Volume 66 , Issue 1 , March 2023 , pp. 142 - 155
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Copyright
© Canadian Mathematical Society, 2022

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Footnotes

The research was supported by a grant from the Simons Foundation, CGM-706591.

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