# Sharp Constants of Approximation Theory. I. Multivariate Bernstein-Nikolskii Type Inequalities

## TL;DR

This paper establishes limit relations for sharp constants in multivariate Bernstein-Nikolskii inequalities, connecting trigonometric polynomials and entire functions of exponential type within convex spectral domains.

## Contribution

It introduces new limit relations between sharp constants in inequalities for multivariate trigonometric polynomials and entire functions of exponential type.

## Key findings

- Derived limit relations for sharp constants in inequalities.
- Connected inequalities for trigonometric polynomials and entire functions.
- Enhanced understanding of multivariate approximation inequalities.

## Abstract

We prove limit relations between the sharp constants in the multivariate Bernstein-Nikolskii type inequalities for trigonometric polynomials and entire functions of exponential type with the spectrum in a centrally symmetric convex body.