On the inequality of different metrics for trigonometric polynomials

Abstract

The article is devoted to the research question of inequalities for different metrics with trigonometric polynomials. The structure of this exploring, its main components and types, as well as its classical approaches are presented in this article. Nikolsky’s inequalities in different metrics are well known for trigonometric polynomials. In this paper, inequalities of different metrics are proved in the Lorentz and Lebesgue spaces for trigonometric polynomials of one variable. A similar result is obtained for trigonometric polynomials of many variables. The article is focused mainly on mathematicians.