Sharp Remez Inequality

S. Tikhonov; P. Yuditskii

doi:10.1007/s00365-019-09473-2

Sharp Remez Inequality

S. Tikhonov, P. Yuditskii

Research output: Contribution to journal › Article › peer-review

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Abstract

Let an algebraic polynomial P_n(ζ) of degree n be such that | P_n(ζ) | ⩽ 1 for ζ∈ E⊂ T and | E| ⩾ 2 π- s. We prove the sharp Remez inequality supζ∈T|Pn(ζ)|⩽Tn(secs4),where T_n 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.

Original language	English
Pages (from-to)	233-246
Number of pages	14
Journal	Constructive Approximation
Volume	52
Issue number	2
DOIs	https://doi.org/10.1007/s00365-019-09473-2
State	Published - 1 Oct 2020
Externally published	Yes

Bibliographical note

Publisher Copyright:
© 2019, Springer Science+Business Media, LLC, part of Springer Nature.

Funding

S. Tikhonov was partially supported by MTM 2017-87409-P, 2017 SGR 358, and the CERCA Programme of the Generalitat de Catalunya. P. Yuditskii was supported by the Austrian Science Fund FWF, Project No: P29363-N32. The authors would like to thank the organizers of the IX Jaen Conference on Approximation Theory (Ubeda, Jaen, Spain), where a part of this work was carried out.

Funders	Funder number
Austrian Science Fund FWF	P29363-N32
Generalitat de Catalunya

Keywords

Comb domains
Sharp Remez inequality
Trigonometric polynomials

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AB - 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.

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Sharp Remez Inequality

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