Markov Inequalities for Weight Functions of Chebyshev Type

Abstract

Denote by η i =cos( iπ/ n), i = 0, ..., n the extreme points of the Chebyshev polynomial T n ( x) = cos( n arc cos x). Let π n be the set of real algebraic polynomials of degree not exceeding n, and let B n be the unit ball in the space π n equipped with the discrete norm | p| n,∞ ≔ max 0 ≤ i ≤ n | p(η i )|. We prove that the unique solutions of the extremal problems max p ∈ B n ∫ 1 −1 [ p ( k + 1) ( x)] 2(1 − x 2) k − 1/2 dx, k = 0, ..., n − 1, and max p ∈ B n ∫ 1 − 1[ p ( k + 2) ( x)] 2(1 − x 2) k − 1/2 dx, k = 0, ..., n − 2, are p( x) = ± T n ( x), and we obtain the extremal values in an explicit form.