Sharp Remez inequality

Abstract

Let an algebraic polynomial $P_n(\zeta )$ of degree $n$ be such that $|P_n(\zeta )|\le 1$ for $\zeta \in E\subset \mathbb{T}$ and $|E|\ge 2\pi -s$. We prove the sharp Remez inequality $$\underset{\zeta \in \mathbb{T}}{\sup }|P_n(\zeta )|\le {\mathfrak{T}}_n\left(\sec \frac{s}{4}\right),$$ where ${\mathfrak{T}}_n$ is the Chebyshev polynomial of degree $n$. The equality holds if and only if $$P_n(e^{iz})=e^{i(nz/2+c_1)}{\mathfrak{T}}_n\left(\sec \frac{s}{4}\cos \frac{z-c_0}{2}\right),c_0,c_1\in \mathbb{R}.$$ This gives the solution of the long-standing problem on the sharp constant in the Remez inequality for trigonometric polynomials.

12 pages, 1 figure; reference added