Sharp Constants of Approximation Theory. VI. Weighted Polynomial Inequalities of Different Metrics on the Multidimensional Cube and Ball

TL;DR

This paper establishes limit equalities for sharp constants in weighted Nikolskii-type inequalities for multivariate polynomials on multidimensional cubes and balls, relating them to entire functions of exponential type.

Contribution

It introduces new limit equalities connecting multivariate polynomial inequalities with entire functions of exponential type in weighted settings.

Findings

Limit equalities between polynomial and entire function constants.

Extension of Nikolskii-type inequalities to weighted multivariate cases.

Unified framework for inequalities on cubes and balls.

Abstract

We prove limit equalities between the sharp constants in weighted Nikolskii-type inequalities for multivariate polynomials on an $m$-dimensional cube and ball and the corresponding constants for entire functions of exponential type.