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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: maxyatts@iupui.edu
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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.

References

Aptekarev, A. I., Sharp constant for rational approximation of analytic functions. Mat. Sb. 193(2002), no. 1, 1–72. English transl. in Math. Sb. 193(1–2):1–72, 2002.CrossRefGoogle Scholar
Aptekarev, A. I. and Yattselev, M., Padé approximants for functions with branch points—strong asymptotics of Nuttall-Stahl polynomials. Acta Math. 215(2015), no. 2, 217–280.CrossRefGoogle Scholar
Astala, K., Iwaniec, T., and Martin, G.. Elliptic partial differential equations and quasiconformal mappings in the plane, Princeton Mathematical Series, 48, Princeton University Press, Princeton, NJ, 2009.Google Scholar
Baratchart, L. and Yattselev, M., Convergent interpolation to Cauchy integrals over analytic arcs with Jacobi-type weights. Int. Math. Res. Not. 2010(2010), no. 22, 4211–4275.Google Scholar
Baxter, G., A convergence equivalence related to polynomials orthogonal on the unit circle. Trans. Amer. Math. Soc. 79(1961), 471–487.CrossRefGoogle Scholar
Bertola, M. and Mo, M. Y., Commuting difference operators, spinor bundles and the asymptotics of orthogonal polynomials with respect to varying complex weights. Adv. Math. 220(2009), 154–218.CrossRefGoogle Scholar
Deift, P., Orthogonal polynomials and random matrices: a riemann-hilbert approach, Courant Lectures in Mathematics, 3, American Mathematical Society, Providence, RI, 2000.Google Scholar
Deift, P., Kriecherbauer, T., McLaughlin, K. T.-R., Venakides, S., and Zhou, X., Strong asymptotics of polynomials orthogonal with respect to exponential weights. Comm. Pure Appl. Math. 52(1999), no. 12, 1491–1552.3.0.CO;2-#>CrossRefGoogle Scholar
Deift, P. and Zhou, X., A steepest descent method for oscillatory Riemann-Hilbert problems. Asymptotics for the mKdV equation. Ann. of Math. 137(1993), 295–370.CrossRefGoogle Scholar
Demengel, F. and Demengel, G., Functional spaces for the theory of elliptic partial differential equations, Universitext, Springer, Berlin, 2012.CrossRefGoogle Scholar
Fokas, A. S., Its, A. R., and Kitaev, A. V., Discrete Panlevé equations and their appearance in quantum gravity. Comm. Math. Phys. 142(1991), no. 2, 313–344.CrossRefGoogle Scholar
Fokas, A. S., Its, A. R., and Kitaev, A. V., The isomonodromy approach to matrix models in 2D quantum gravitation. Comm. Math. Phys. 147(1992), no. 2, 395–430.CrossRefGoogle Scholar
Gakhov, F. D., Boundary value problems, Dover Publications, Inc., New York, 1990.Google Scholar
Kilgore, T., On the simultaneous approximation of functions and their derivatives . In: Applied mathematics reviews. Vol. 1, World Scientific Publishing, River Edge, NJ, 2000, pp. 69–118.CrossRefGoogle Scholar
Kuijlaars, A. B., McLaughlin, K. T.-R., Van Assche, W., and Vanlessen, M., The Riemann-Hilbert approach to strong asymptotics for orthogonal polynomials on [−1,1]. Adv. Math. 188(2004), no. 2, 337–398.CrossRefGoogle Scholar
McLaughlin, K. T.-R. and Miller, P. D., The $\bar{\partial }$ steepest descent method for orthogonal polynomials on the real line with varying weights . Int. Math. Res. Not. IMRN 2008(2008), rnn075.CrossRefGoogle Scholar
Simon, B., Orthogonal polynomials on the unit circle . Vol. I, II, Colloquium Publications, 54, American Mathematical Society, Providence, RI, 2005.Google Scholar
Szegő, G., Orthogonal polynomials, Colloquium Publications, 23, American Mathematical Society, Providence, RI, 1999.Google Scholar