An asymptotic formula for polynomials orthonormal with respect to a varying weight
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- by A. V. Komlov and S. P. Suetin
- Trans. Moscow Math. Soc. 2012, 139-159
- DOI: https://doi.org/10.1090/S0077-1554-2013-00204-6
- Published electronically: March 21, 2013
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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 $\alpha _{n}(n)e^{[b]{-nw^{}_Q}}$ oscillates as $n\to \infty$ and, in a typical case, fills an interval (here $w_Q$ is the equilibrium constant in the external field $Q$).References
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Bibliographic 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
- MR Author ID: 190281
- Email: suetin@mi.ras.ru
- Published electronically: March 21, 2013
- © Copyright 2013 American Mathematical Society
- Journal: Trans. Moscow Math. Soc. 2012, 139-159
- MSC (2010): Primary 42C05; Secondary 33C45, 33C50, 33D45
- DOI: https://doi.org/10.1090/S0077-1554-2013-00204-6
- MathSciNet review: 3184971