On interrelation of Nikolskii Constants for trigonometric polynomials and entire functions of exponential type

Abstract

For $0<p<\infty$, we investigate the interrelation between the Nikolskii
constant for trigonometric polynomials of order at most $n$
\[
\mathcal{C}(n,p)=\sup_{T_{n}\ne 0}\frac{\|T_{n}\|_{\infty}}{\|T_{n}\|_{p}}
\]
and the Nikolskii constant for entire functions of exponential type at most~$1$
\[
\mathcal{L}(p)=\sup_{f\ne 0}\frac{\|f\|_{\infty}}{\|f\|_{p}}.
\]

Recently E.~Levin and D.~Lubinsky have proved that
\[
\mathcal{C}(n,p)=\mathcal{L}(p)n^{1/p}(1+o(1)),\quad n\to \infty.
\]
M.~Ganzburg and S.~Tikhonov have extend this result on the case of
Nikolskii--Bernstein constants.

We prove inequalities
\[
n^{1/p}\mathcal{L}(p)\le \mathcal{C}(n,p)\le (n+\lceil
p^{-1}\rceil)^{1/p}\mathcal{L}(p),\quad n\in \mathbb{Z}_{+},\quad 0<p<\infty,
\]
which improve the result of Levin and Lubinsky. The proof follows our old
approach based on properties of the integral Fejer kernel. Using this approach
we proved earlier estimates for $p=1$
\[
n\mathcal{L}(1)\le \mathcal{C}(n,1)\le (n+1)\mathcal{L}(1).
\]

Using such inequalities, we can estimate the constant $\mathcal{L}(p)$ solving
approximately $\mathcal{C}(n,p)$ for large $n$. To do this we use recent
results of V.~Arestov and M.~Deikalova, who expressed the Nikolskii constant
$\mathcal{C}(n,p)$ using the algebraic polynomial $\rho_{n}$ that deviates
least from zero in the space $L^{p}$ on the segment $[-1,1]$ with the weight
$(1-t)v(t)$, where $v(t)=(1-t^{2})^{-1/2}$ is the Chebyshev weight. As
consequence, we refine estimates of the Nikolskii constant $\mathcal{L}(1)$ and
find~that
\[
1.081<2\pi \mathcal{L}(1)<1.082.
\]
To compare previous estimates were $1.081<2\pi \mathcal{L}(1)<1.098$.