Asymptotics of sharp constants of Markov-Bernstein inequalities in integral norm with Jacobi weight
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- by A. I. Aptekarev, A. Draux, V. A. Kalyagin and D. N. Tulyakov PDF
- Proc. Amer. Math. Soc. 143 (2015), 3847-3862 Request permission
Abstract:
The classical A. Markov inequality establishes a relation between the maximum modulus or the $L^{\infty }\left ([-1,1]\right )$ norm of a polynomial $Q_{n}$ and of its derivative: $\|Q’_{n}\|\leqslant M_{n} n^{2}\|Q_{n}\|$, where the constant $M_{n}=1$ is sharp. The limiting behavior of the sharp constants $M_{n}$ for this inequality, considered in the space $L^{2}\left ([-1,1], w^{(\alpha ,\beta )}\right )$ with respect to the classical Jacobi weight $w^{(\alpha ,\beta )}(x):=(1-x)^{\alpha }(x+1)^{\beta }$, is studied. We prove that, under the condition $|\alpha - \beta | < 4$, the limit is $\lim _{n \to \infty } M_{n} = 1/(2 j_{\nu })$ where $j_{\nu }$ is the smallest zero of the Bessel function $J_{\nu }(x)$ and $2 \nu =\mbox {min}(\alpha , \beta ) - 1$.References
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Additional Information
- A. I. Aptekarev
- Affiliation: Keldysh Institute of Applied Mathematics, Russian Academy of Sciences, Moscow, Russia and Moscow State University, Moscow, Russia
- MR Author ID: 192572
- Email: aptekaa@keldysh.ru
- A. Draux
- Affiliation: INSA, Rouen, France
- Email: andre.draux@insa-rouen.fr
- V. A. Kalyagin
- Affiliation: National Research University Higher School of Economics, Nizhny Novgorod, Russia
- Email: vkalyagin@hse.ru
- D. N. Tulyakov
- Affiliation: Keldysh Institute of Applied Mathematics, Russian Academy of Sciences, Moscow, Russia
- MR Author ID: 632175
- Email: dntulyakov@gmail.com
- Received by editor(s): June 11, 2013
- Received by editor(s) in revised form: February 24, 2014, and April 18, 2014
- Published electronically: May 4, 2015
- Additional Notes: The first and fourth authors were partly supported by the program N1 of DMS RAS and grants RFBR-13-01-12430, RFBR-14-01-00604. The third author was partly supported by the Scientific Schools program - 2900.2014.1. The paper was finished while the first author visited INSA, Rouen, France.
- Communicated by: W. Van Assche
- © Copyright 2015 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 143 (2015), 3847-3862
- MSC (2010): Primary 41A20.21, 33C45, 42C05; Secondary 47B99, 30B70
- DOI: https://doi.org/10.1090/proc/12535
- MathSciNet review: 3359576