Improvement of one inequality for algebraic polynomials

• Date: 2009
• Main Authors: Nesterenko, A. N., Tymoshkevych, T. D., Chaikovs'kyi, A. V., Нестеренко, О. Н., Тимошкевич, Т. Д., Чайковський, А. В.
• Format: Article
• Language: Ukrainian; English
• Published: Institute of Mathematics, NAS of Ukraine 2009
• Online Access: https://umj.imath.kiev.ua/index.php/umj/article/view/3015
• Journal Title: Ukrains’kyi Matematychnyi Zhurnal

We prove that the inequality $||g(⋅/n)||_{L_1[−1,1]}||P_{n+k}||_{L_1[−1,1]} ≤ 2||gP_{n+k}||_{L_1[−1,1]}$, where $g : [-1, 1]→ℝ$ is a monotone odd function and $P_{n+k}$ is an algebraic polynomial of degree not higher than $n + k$, is true for all natural $n$ for $k = 0$ and all natural $n ≥ 2$ for $...