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

TL;DR

This paper establishes an asymptotic equality for sharp constants in multivariate polynomial inequalities with given Newton polyhedra, linking them to entire functions of exponential type as the polyhedron scales infinitely.

Contribution

It proves a limit equality connecting polynomial inequality constants with those for entire functions, advancing understanding of asymptotic behavior in approximation theory.

Findings

Established asymptotic equality for sharp constants as polyhedron scale increases

Linked multivariate polynomial inequalities to entire functions of exponential type

Provided theoretical foundation for approximation bounds in multivariate settings

Abstract

Let $V\subset\R^m$ 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 $V$.