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

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.