Sharp Remez Inequality

Abstract

Let an algebraic polynomial $$P_n(\zeta )$$Pn(ζ)of degree n be such that $$|P_n(\zeta )|\leqslant 1$$|Pn(ζ)|⩽1for $$\zeta \in E\subset \mathbb {T}$$ζ∈E⊂Tand $$|E|\geqslant 2\pi -s$$|E|⩾2π-s. We prove the sharp Remez inequality $$\begin{aligned} \sup _{\zeta \in \mathbb {T}}|P_n(\zeta )|\leqslant {\mathfrak {T}}_{n}\left( \sec \frac{s}{4}\right) , \end{aligned}$$supζ∈T|Pn(ζ)|⩽Tnsecs4,where $${\mathfrak {T}}_{n}$$Tnis the Chebyshev polynomial of degree n. The equality holds if and only if $$\begin{aligned} P_n(e^{iz})=e^{i(nz/2+c_1)}{\mathfrak {T}}_n\left( \sec \frac{s}{4}\cos \frac{z-c_0}{2}\right) , \quad c_0,c_1\in {\mathbb {R}}. \end{aligned}$$Pn(eiz)=ei(nz/2+c1)Tnsecs4cosz-c02,c0,c1∈R.This gives the solution of the long-standing problem on the sharp constant in the Remez inequality for trigonometric polynomials.