Sharp Remez Inequality

S. Tikhonov, P. Yuditskii

    Research output: Contribution to journalArticleResearch

    Abstract

    © 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.
    Original languageEnglish
    JournalConstructive Approximation
    DOIs
    Publication statusPublished - 1 Jan 2019

    Keywords

    • Comb domains
    • Sharp Remez inequality
    • Trigonometric polynomials

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