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

Волошина, В. О. (2022). On exact constant in Dzyadyk inequality for the derivative of an algebraic polynomial. Bulletin of Taras Shevchenko National University of Kyiv. Physics and Mathematics(1), 34–37. https://doi.org/10.17721/1812-5409.2022/1.3

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.
Pages of the article in the issue: 34 - 37
Language of the article: Ukrainian