# Sharp Constants of Approximation Theory. II. Invariance Theorems and Certain Multivariate Inequalities of Different Metrics

## TL;DR

This paper establishes invariance theorems for multivariate inequalities involving sharp constants, connecting polynomial inequalities on spheres and balls with entire functions, advancing approximation theory in multiple metrics.

## Contribution

It introduces invariance theorems that relate sharp constants in multivariate polynomial inequalities to entire functions, expanding understanding of approximation bounds across different metrics.

## Key findings

- Proved invariance theorems for inequalities of different metrics.
- Established limit relations between sharp constants for polynomials and entire functions.
- Discussed relations in univariate weighted spaces.

## Abstract

We prove invariance theorems for general inequalities of different metrics and apply them to limit relations between the sharp constants in the multivariate Markov-Bernstein-Nikolskii type inequalities with the polyharmonic operator for algebraic polynomials on the unit sphere and the unit ball in $\R^m$ and the corresponding constants for entire functions of spherical type on $\R^m$. Certain relations in the univariate weighted spaces are discussed as well.