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On the asymptotics of sharp constants in Markov-Bernstein inequalities in integral metrics with classical weight. (English. Russian original) Zbl 0957.41008

Russ. Math. Surv. 55, No. 1, 163-165 (2000); translation from Usp. Mat. Nauk 55, No. 1, 173-174 (2000).
Given a Banach space \(X\) of continuous functions containing the polynomials the Markov-Bernstein inequalities have the form \(\|P'\|\leq M_n\|P\|\) \((\deg P\leq n)\). In this paper, the authors obtain asymptotics for the best constants \(M_n\) in the Markov-Bernstein inequalities in two cases of weighted function spaces \(X=L^2(I,\omega)\), namely for \(I=[0,\infty)\) and the weight function \(\omega(x)= x^\alpha \exp(-x)\) with \(\alpha>-1\) (generalized Laguerre weight) and for \(I=[-1,1]\) and the weight function \(\omega(x)=(1-x^2)^\gamma\) with \(\gamma>-1\) (Gegenbauer weight). It turns out that, in the first case \(M_n={n \over x_1}[1+ O(1)]\) and in the second case \(M_n={n^2 \over 2x_1}[1+ O(1)]\), where \(x_1\) is the zero nearest to the origin of the Bessel-function \(J_\beta\) with \(\beta= (\alpha-1)/2\) or \(\beta= (r-1)/2\), respectively.

MSC:

41A17 Inequalities in approximation (Bernstein, Jackson, Nikol’skiĭ-type inequalities)
41A44 Best constants in approximation theory
46E15 Banach spaces of continuous, differentiable or analytic functions
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