On Sharp Constants in Bernstein–Nikolskii Inequalities

Michael I. Ganzburg, Sergey Yu Tikhonov

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    19 Citations (Scopus)

    Abstract

    © 2017, Springer Science+Business Media New York. Let Tn be the set of all trigonometric polynomials of degree at most n, and let B1 be the set of all entire functions of exponential type at most 1. We discuss limit relations between the sharp constants in the Bernstein–Nikolskii inequalities defined by Ap,q,n(s):=n-s-1/p+1/qsupT∈Tn\{0}‖T(s)‖Lq([-π,π])‖T‖Lp([-π,π]),Dp,q(s):=supf∈(B1∩Lp(R))\{0}‖f(s)‖Lq(R)‖f‖Lp(R),where 0 < p< q≤ ∞ and s=0,1,…. We prove that Dp,q(s)≤lim infn→∞Ap,q,n(s),Dp,∞(s)=limn→∞Ap,∞,n(s).
    Original languageEnglish
    Pages (from-to)449-466
    JournalConstructive Approximation
    Volume45
    Issue number3
    DOIs
    Publication statusPublished - 1 Jun 2017

    Keywords

    • Bernstein–Nikolskii inequality
    • Entire functions of exponential type
    • Levitan’s polynomials
    • Sharp constants
    • Trigonometric polynomials

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