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Transactions of the Moscow Mathematical Society
Transactions of the Moscow Mathematical Society
ISSN 1547-738X(online) ISSN 0077-1554(print)

   

 

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


Authors: A. V. Komlov and S. P. Suetin
Translated by:
Original publication: Trudy Moskovskogo Matematicheskogo Obshchestva.
Journal: Trans. Moscow Math. Soc. 2012, 139-159
MSC (2010): Primary 42C05; Secondary 33C45, 33C50, 33D45
Published electronically: March 21, 2013
MathSciNet review: 3184971
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Abstract | References | Similar Articles | Additional Information

Abstract: We obtain a strong asymptotic formula for the leading coefficient $ \alpha _{n}(n)$ of a degree $ n$ polynomial $ q_{n}(z;n)$ orthonormal on a system of intervals on the real line with respect to a varying weight. The weight depends on $ n$ as $ e^{-2nQ(x)}$, where $ Q(x)$ is a polynomial and corresponds to the ``hard-edge case''. The formula in Theorem 1 is quite similar to Widom's classical formula for a weight independent of $ n$. In some sense, Widom's formulas are still true for a varying weight and are thus universal. As a consequence of the asymptotic formula we have that $ \alpha _{n}(n)e^{\smash [b]{-nw^{}_Q}}$ oscillates as $ n\to \infty $ and, in a typical case, fills an interval (here $ w_Q$ is the equilibrium constant in the external field $ Q$).


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Additional Information

A. V. Komlov
Affiliation: Steklov Mathematical Institute of the Russian Academy of Sciences
Email: komlov@mi.ras.ru

S. P. Suetin
Affiliation: Steklov Mathematical Institute of the Russian Academy of Sciences
Email: suetin@mi.ras.ru

DOI: http://dx.doi.org/10.1090/S0077-1554-2013-00204-6
Keywords: Varying weight, orthonormal polynomials, strong asymptotics, equilibrium distributions.
Published electronically: March 21, 2013
Article copyright: © Copyright 2013 American Mathematical Society