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Статья

Asymptotics of sharp constants of markov-bernstein inequalities in integral norm with jacobi weight

Proceedings of the American Mathematical Society. 2015. Vol. 143. No. 9. P. 3847-3862.
Aptekarev A., Draux A., Kalyagin V. A., Tulyakov D.

The classical A. Markov inequality establishes a relation between the maximum modulus or the L∞ ([−1, 1]) norm of a polynomial Qn and of its derivative: ||Qʹn|| ≤Mnn2||Qn|| where the constant Mn = 1 is sharp. The limiting behavior of the sharp constants Mn for this inequality, considered in the space L2 ([−1, 1], w(α,β)(x) 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 limn→∞Mn = 1/(2jν) where jν is the smallest zero of the Bessel function Jν(x) and 2ν = min(α, β) − 1. © 2015, American Mathematical Society.