Some inequalities for derivatives of trigonometric and algebraic polynomials (Q1923651)

Let \(W\) be a \(2\pi\)-periodic even weight of class \(A_p\) and \(T_n:\mathbb{R}\to \mathbb{R}\) a trigonometric polynomial of degree at most \(n\). Then for all \(\ell\in \mathbb{N}\), \[ |T_n^{(\ell)} |_{p,W}\leq 3\ell(2n)^\ell \omega \bigl(T_n; {\textstyle {\pi\over4n}} \bigr)_{p,W}. \] Here \(|.|_{p,W}\) stands for the norm in the weighted Lebesgue space \(L^p_W((0,2\pi))\) and the modulus of continuity \(\omega(T_n;\delta)\), \(\delta\geq 0\), is given by \[ \omega(T_n; \delta)_{p,W}= \sup_{|h-k|\leq \delta}|T_n(.+h)- T_n(.+k) |_{p,W}. \] This is one of the main results of the paper which generalizes to higher derivatives Theorem 1 of \textit{K. Balázs} and \textit{Th. Kolgore} [J. Approximation Theory 82, 274-286 (1995; Zbl 0840.42002)]. Applications to algebraic polynomials are given. The authors also prove an inequality of Brudnyi in terms of \(r\)th order moduli of continuity \(\omega_r\).