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

D. V. Gorbachev ; I. A. Martyanov

Čebyševskij sbornik, Tome 19 (2018) no. 2, pp. 80-89

Résumé

For $0$, 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 $$ n^{1/p}\mathcal{L}(p)\le \mathcal{C}(n,p)\le (n+\lceil p^{-1}\rceil)^{1/p}\mathcal{L}(p),\quad n\in \mathbb{Z}_{+},\quad 0\infty, $$ 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 $$ 1.0812\pi \mathcal{L}(1)1.082. $$ To compare previous estimates were $1.0812\pi \mathcal{L}(1)1.098$.

Détail

Keywords: trigonometric polynomial, entire function of exponential type, Nikolskii constant, Chebyshev weight.