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Asymptotics of sharp constants of Markov-Bernstein inequalities in integral norm with Jacobi weight

Authors: A. I. Aptekarev, A. Draux, V. A. Kalyagin and D. N. Tulyakov
Journal: Proc. Amer. Math. Soc. 143 (2015), 3847-3862
MSC (2010): Primary 41A20, 33C45, 42C05; Secondary 47B99, 30B70
Published electronically: May 4, 2015
MathSciNet review: 3359576
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Abstract | References | Similar Articles | Additional Information

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: $ \Vert Q'_{n}\Vert\leqslant M_{n} n^{2}\Vert Q_{n}\Vert$, 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 $ \vert\alpha - \beta \vert < 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$.

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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

A. Draux
Affiliation: INSA, Rouen, France

V. A. Kalyagin
Affiliation: National Research University Higher School of Economics, Nizhny Novgorod, Russia

D. N. Tulyakov
Affiliation: Keldysh Institute of Applied Mathematics, Russian Academy of Sciences, Moscow, Russia

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
Article copyright: © Copyright 2015 American Mathematical Society

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