TY - JOUR

T1 - Sharp Remez Inequality

AU - Tikhonov, S.

AU - Yuditskii, P.

PY - 2019/1/1

Y1 - 2019/1/1

N2 - © 2019, Springer Science+Business Media, LLC, part of Springer Nature. 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.

AB - © 2019, Springer Science+Business Media, LLC, part of Springer Nature. 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.

KW - Comb domains

KW - Sharp Remez inequality

KW - Trigonometric polynomials

UR - http://www.mendeley.com/research/sharp-remez-inequality

U2 - 10.1007/s00365-019-09473-2

DO - 10.1007/s00365-019-09473-2

M3 - Article

JO - Constructive Approximation

JF - Constructive Approximation

SN - 0176-4276

ER -