On exact constant in Dzyadyk inequality for the derivative of an algebraic polynomial

Abstract

Bernstein inequality made it possible to obtain a constructive characterization of the approximation of periodic functions by trigonometric polynomials T_n of degree n. Instead, the corollary of this inequality for algebraic polynomials P_n of degree n, namely, the inequality $||? P_n'|| ? n ||P_n||$, where $? · ? := ? · ?_[?1,1]$ and $?(x) := \sqrt{1-x^2}$, does not solve the problem obtaining a constructive characterization of the approximation of continuous functions on a segment by algebraic polynomials. Markov inequality $||P_n'|| ? n^2 ||P_n||$ does not solve this problem as well. Moreover, even the corollary $||?_n P_n'|| ? 2n ||P_n||$, where $?_n(x) := \sqrt{1-x^2+1/n^2}$ of Bernstein and Markov inequalities is not enough. This problem, like a number of other theoretical and practical problems, is solved by Dzyadyk inequality $|| P_n' ?_n^{1-k} || ? c(s) n|| P_n ?_n^{-s} ||,$ valid for each s ? R. In contrast to the Bernstein and Markov inequalities, the exact constant in the Dzyadyk inequality is unknown for all s ? R, whereas the asymptotically exact constant for natural s is known: c(s) = 1 + s + s^2; and for n ? 2s, s ? N, even the exact constant is known. In our note, this result is extended to the case s ? n < 2s.