A Sharp Remez Inequality for Trigonometric Polynomials

We obtain a sharp Remez inequality for the trigonometric polynomial T _n of degree n on 0,2π): $$\|T_n \|_{L_\infty(0,2\pi))} \le \biggl(1+2\tan^2 \frac{n\beta}{4m} \biggr) { \|T_n \|_{L_\infty (0,2\pi) \setminus B )}}, $$ where $\frac{2\pi}{m}$ is the minimal period of T _n and $|B|=\beta<\frac {2\pi m}{n}$ is a measurable subset of $\mathbb {T}$ . In particular, this gives the asymptotics of the sharp constant in the Remez inequality: for a fixed n, $$\mathcal{C}(n, \beta)=1+ \frac{(n\beta)^2}{8} +O \bigl(\beta^4\bigr), \quad\beta \to0, $$ where $$\mathcal{C}(n,\beta):= \sup_{|B|=\beta}\sup_{T_n} \frac{ \|T_n \|_{L_\infty(0,2\pi ))}}{ \|T_n \|_{L_\infty (0,2\pi) \setminus B )}}. $$ We also obtain sharp Nikol’skii’s inequalities for the Lorentz spaces and net spaces. Multidimensional variants of Remez and Nikol’skii’s inequalities are investigated.