Certain inequalities in various metrics for trigonometric polynomials and their derivatives

V. I. Ivanov

Matematičeskie zametki, Tome 18 (1975) no. 4, pp. 489-498.

Résumé

We establish for $0$ the analog of the Bernstein–Zygmund inequality for the derivative of a trigonometric polynomial $$ \int_{-\pi}^\pi|t_n'(x)|^pdx\leqslant c_pn^p\int_{-\pi}^\pi|t_n(x)|^pdx. $$ We prove weighted inequalities, exact in the sense of order, for trigonometric polynomials and their derivatives in various integral metrics with exponents $0$, $q\leqslant\infty$.