Sharp Constants of Approximation Theory. V. An Asymptotic Equality Related to Polynomials with Given Newton Polyhedra

Abstract

Let $V\subset {\mathbb{R}}^m$ be a convex body, symmetric about all coordinate hyperplanes, and let $\PP_{aV},\, a\ge 0$, be a set of all algebraic polynomials whose Newton polyhedra are subsets of $aV$. We prove a limit equality as $a\to \iy$ between the sharp constant in the multivariate Markov-Bernstein-Nikolskii type inequalities for polynomials from $\PP_{aV}$ and the corresponding constant for entire functions of exponential type with the spectrum in $V$.

arXiv comment22 pages. arXiv admin note: text overlap with arXiv:1902.10215