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 Mat. Sb., 2014, Volume 205, Number 9, Pages 121–144 (Mi msb8360)

An asymptotic formula for polynomials orthonormal with respect to a varying weight. II

A. V. Komlov*, S. P. Suetin

Steklov Mathematical Institute of Russian Academy of Sciences

Abstract: This paper gives a proof of the theorem announced by the authors in the preceding paper with the same title. The theorem states that asymptotically the behaviour of the polynomials which are orthonormal with respect to the varying weight $e^{-2nQ(x)}p_g(x)/\sqrt{\prod_{j=1}^{2p}(x-e_j)}$ coincides with the asymptotic behaviour of the Nuttall psi-function, which solves a special boundary-value problem on the relevant hyperelliptic Riemann surface of genus $g=p-1$. Here $e_1<e_2<…<e_{2p}$, $Q(x)=x^{2m}+\dotsb$ is a monic polynomial of even degree $2m$ and $p_g$ is a certain auxiliary polynomial of degree $p-1$.
Bibliography: 23 titles.

Keywords: varying weight, orthonormal polynomials, strong asymptotics, uniform distributions.

 Funding Agency Grant Number Russian Foundation for Basic Research 13-01-12430-îôè-ì213-01-00622-à14-01-31281-ìîë-à Ministry of Education and Science of the Russian Federation ÍØ-2900.2014.1

* Author to whom correspondence should be addressed

DOI: https://doi.org/10.4213/sm8360

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English version:
Sbornik: Mathematics, 2014, 205:9, 1334–1356

Bibliographic databases:

UDC: 517.53

Citation: A. V. Komlov, S. P. Suetin, “An asymptotic formula for polynomials orthonormal with respect to a varying weight. II”, Mat. Sb., 205:9 (2014), 121–144; Sb. Math., 205:9 (2014), 1334–1356

Citation in format AMSBIB
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• http://mi.mathnet.ru/eng/msb8360
• https://doi.org/10.4213/sm8360
• http://mi.mathnet.ru/eng/msb/v205/i9/p121

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This publication is cited in the following articles:
1. V. I. Buslaev, S. P. Suetin, “On equilibrium problems related to the distribution of zeros of the Hermite–Padé polynomials”, Proc. Steklov Inst. Math., 290:1 (2015), 256–263
2. S. P. Suetin, “Distribution of the zeros of Padé polynomials and analytic continuation”, Russian Math. Surveys, 70:5 (2015), 901–951
3. N. R. Ikonomov, R. K. Kovacheva, S. P. Suetin, “Nuttall's integral equation and Bernshtein's asymptotic formula for a complex weight”, Izv. Math., 79:6 (2015), 1215–1234
4. E. Levin, D. Lubinsky, Bounds and asymptotics for orthogonal polynomials for varying weights, Springerbriefs in Mathematics, Springer, 2018
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