Sharp constants of approximation theory. V. An asymptotic equality related to polynomials with given Newton polyhedra

Abstract

Let V⊂Rm be a convex body, symmetric about all coordinate hyperplanes, and let PaV,a≥0, be a set of all algebraic polynomials whose Newton polyhedra are subsets of aV. We prove a limit equality as a→∞ between the sharp constant in the multivariate Markov-Bernstein-Nikolskii type inequalities for polynomials from PaV and the corresponding constant for entire functions of exponential type with the spectrum in V.