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 Chebyshevskii Sb., 2019, Volume 20, Issue 3, Pages 143–153 (Mi cheb804)

Interrelation between Nikolskii–Bernstein constants for trigonometric polynomials and entire functions of exponential type

D. V. Gorbachev, I. A. Martyanov

Tula State University (Tula)

Abstract: Let $0<p\le \infty$, $\mathcal{C}(n;p;r)=\sup_{T}\frac{\|T^{(r)}\|_{L^{\infty}[0,2\pi)}}{\|T\|_{L^{p}[0,2\pi)}}$ and $\mathcal{L}(p;r)=\sup_{F}\frac{\|F^{(r)}\|_{L^{\infty}(\mathbb{R})}}{\|F\|_{L^{p}(\mathbb{R})}}$ be the sharp Nikolskii–Bernstein constants for $r$-th derivatives of trigonometric polynomials of degree $n$ and entire functions of exponential type $1$ respectively. Recently E. Levin and D. Lubinsky have proved that for the Nikolskii constants
$$\mathcal{C}(n;p;0)=n^{1/p}\mathcal{L}(p;0)(1+o(1)),\quad n\to \infty.$$
M. Ganzburg and S. Tikhonov generalized this result to the case of Nikolskii–Bernstein constants:
$$\mathcal{C}(n;p;r)=n^{r+1/p}\mathcal{L}(p;r)(1+o(1)),\quad n\to \infty.$$
They also showed the existence of the extremal polynomial $\tilde{T}_{n,r}$ and the function $\tilde{F}_{r}$ in this problem, respectively. Earlier, we gave more precise boundaries in the Levin–Lubinsky-type result, proving that for all $p$ and $n$
$$n^{1/p}\mathcal{L}(p;0)\le \mathcal{C}(n;p;0)\le (n+\lceil 1/p\rceil)^{1/p}\mathcal{L}(p;0).$$
Here we establish close facts for the case of Nikolskii–Bernstein constants, which also imply the asymptotic Ganzburg–Tikhonov equality. The results are stated in terms of extremal functions $\tilde{T}_{n,r}$, $\tilde{F}_{r}$ and the Taylor coefficients of a kernel of type Jackson–Fejer $(\frac{\sin \pi x}{\pi x})^{2s}$. We implicitly use Levitan-type polynomials arising from the Poisson summation formula. We formulate one hypothesis about the signs of the Taylor coefficients of the extremal functions.

Keywords: trigonometric polynomial, entire function of exponential type, Nikolskii–Bernstein constant, Jackson–Fejer kernel, Levitan polynomials.

 Funding Agency Grant Number Russian Science Foundation 18-11-00199 This Research was performed by a grant of Russian Science Foundation (project 18-11-00199).

DOI: https://doi.org/10.22405/2226-8383-2018-20-3-143-153

Full text: PDF file (698 kB)

UDC: 517.5
Accepted:12.11.2019

Citation: D. V. Gorbachev, I. A. Martyanov, “Interrelation between Nikolskii–Bernstein constants for trigonometric polynomials and entire functions of exponential type”, Chebyshevskii Sb., 20:3 (2019), 143–153

Citation in format AMSBIB
\Bibitem{GorMar19} \by D.~V.~Gorbachev, I.~A.~Martyanov \paper Interrelation between Nikolskii--Bernstein constants for~trigonometric polynomials and entire functions of~exponential~type \jour Chebyshevskii Sb. \yr 2019 \vol 20 \issue 3 \pages 143--153 \mathnet{http://mi.mathnet.ru/cheb804} \crossref{https://doi.org/10.22405/2226-8383-2018-20-3-143-153}