Sharp Bernstein--Nikolskii inequalities for polynomials and entire functions of exponential type

Abstract

We survey Bernstein--Nikolskii inequalities for polynomials and entire functions of exponential type, with particular emphasis on sharp constants and extremal functions. We discuss both classical results and recent developments, including limit relations between Nikolskii constants for polynomials and entire functions of exponential type, weighted and multidimensional versions, and inequalities on constrained classes of functions. Special attention is given to the difficult case $p=1$ and to methods based on reproducing kernels, duality, and extremal approximation. We also describe connections of sharp Bernstein--Nikolskii inequalities with approximation theory, the Riemann zeta function, spherical designs, Tur\'an--Delsarte type problems, and sphere packing. Several open problems concerning sharp constants, extremal functions, and multidimensional extensions are formulated.