On Sharp Constants in Bernstein–Nikolskii Inequalities

Abstract

Let ${\mathcal T}_n$ be the set of all trigonometric polynomials of degree at most n, and let $B_1$ be the set of all entire functions of exponential type at most 1. We discuss limit relations between the sharp constants in the Bernstein–Nikolskii inequalities defined by

$$\begin{aligned}&A_{p,q,n}^{(s)}:=n^{-s-1/p+1/q}\sup _{T\in {\mathcal T}_n{\setminus }\{0\}}\frac{\Vert T^{(s)}\Vert _{L_q([-\pi ,\pi ])}}{\Vert T\Vert _{L_p([-\pi ,\pi ])}},\quad \\&D_{p,q}^{(s)}:= \sup _{f\in (B_1\cap L_p({\mathbb R})){\setminus }\{0\}}\frac{\Vert f^{(s)}\Vert _{L_q({\mathbb R})}}{\Vert f\Vert _{L_p({\mathbb R})}}, \end{aligned}$$

where $0<p<q\le {\infty }$ and $s=0,\,1,\ldots .$ We prove that

$$\begin{aligned} D_{p,q}^{(s)}\le \liminf _{n\rightarrow {\infty }}A_{p,q,n}^{(s)},\qquad D_{p,{\infty }}^{(s)}= \lim _{n\rightarrow {\infty }}A_{p,{\infty },n}^{(s)}. \end{aligned}$$