|
This article is cited in 11 scientific papers (total in 11 papers)
On interrelation of Nikolskii Constants for trigonometric polynomials and entire functions of exponential type
D. V. Gorbachev, I. A. Martyanov Tula State University
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 $1.081<2\pi \mathcal{L}(1)<1.098$.
Keywords:
trigonometric polynomial, entire function of exponential type, Nikolskii constant, Chebyshev weight.
Received: 05.06.2018 Accepted: 17.08.2018
Citation:
D. V. Gorbachev, I. A. Martyanov, “On interrelation of Nikolskii Constants for trigonometric polynomials and entire functions of exponential type”, Chebyshevskii Sb., 19:2 (2018), 80–89
Linking options:
https://www.mathnet.ru/eng/cheb640 https://www.mathnet.ru/eng/cheb/v19/i2/p80
|
Statistics & downloads: |
Abstract page: | 147 | Full-text PDF : | 41 | References: | 14 |
|