Bernstein-Szegő inequality for the Riesz derivative of trigonometric polynomials in $L_p$-spaces, $0\leqslant p\leqslant\infty$, with classical value of the sharp constant

Abstract

The Bernstein-Szegő inequality for the Weyl derivative of real order $\alpha\geqslant 0$ of trigonometric polynomials of degree $n$ is considered. The aim is to find values of the parameters for which the sharp constant in this inequality is equal to $n^\alpha$ (the classical value) in all $L_p$-spaces, $0\leqslant p\leqslant\infty$. The set of all such $\alpha$ is described for some important particular cases of the Weyl-Szegő derivative, namely, for the Riesz derivative and for the conjugate Riesz derivative, for all $n\in\mathbb N$. Bibliography: 22 titles.