Bernstein-Nikolskii Inequality: Optimality with Respect to the Smoothness Parameter

Abstract

In this paper, we study the form of the constant $C$ in the Bernstein--Nikolskii inequalities $\Vert f^{(s)}{\Vert }_q\lesssim C(s,p,q){\Vert f\Vert }_p,0<p<q\le \infty$, for trigonometric polynomials and entire functions of exponential type. We obtain the optimal behavior of the constant with respect to the smoothness parameter $s$.

arXiv comment22 pages