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

2014/05/01 by A. I. Aptekarev, A. Draux, Aptekarev, A. I. +5
Mathematics · #33C45 #41A20.21 #42C05 #Classical Analysis and ODEs (math.CA) #FOS: Mathematics #math.CA #msc:33C45 #msc:41A20.21 #msc:42C05

paper · pdf · doi:10.48550/arxiv.1405.0167

12 pages, biblio 15 sources

arxiv created 2014/05/01 · arxiv updated 2014/05/02

Abstract

The classical A. Markov inequality establishes a relation between the maximum modulus or the L^∞([-1,1]) norm of a polynomial Q_n and of its derivative: ‖Q'_n‖\leqslant M_n n²‖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 L²([-1,1], w^(α,β)) with respect to the classical Jacobi weight w^(α,β)(x):=(1-x)^α(x+1)^β, is studied. We prove that, under the condition |α- β| < 4 , the limit is lim_{n → ∞} M_n = 1/(2 j_ν) where j_ν is the smallest zero of the Bessel function J_ν(x) and 2 ν= min(α, β) - 1.

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