Sharp Remez Inequality

Abstract

Let an algebraic polynomial Pn(ζ) of degree n be such that | Pn(ζ) | ⩽ 1 for ζ∈ E⊂ T and | E| ⩾ 2 π- s. We prove the sharp Remez inequality supζ∈T|Pn(ζ)|⩽Tn(secs4),where Tn is the Chebyshev polynomial of degree n. The equality holds if and only if Pn(eiz)=ei(nz/2+c1)Tn(secs4cosz-c02),c0,c1∈R.This gives the solution of the long-standing problem on the sharp constant in the Remez inequality for trigonometric polynomials. © 2019, Springer Science+Business Media, LLC, part of Springer Nature.