# On approximations by trigonometric polynomials of classes of functions defined by moduli of smoothness

## TL;DR

This paper characterizes Nikol'skii-Besov function classes using moduli of smoothness and Fourier coefficients, providing necessary and sufficient conditions for membership based on recent inequalities.

## Contribution

It offers a new characterization of function classes via series over moduli of smoothness and Fourier coefficients, advancing understanding of function approximation.

## Key findings

- Characterization of Nikol'skii-Besov classes through moduli of smoothness.
- Necessary and sufficient conditions using Fourier coefficients.
- Application of reverse Copson- and Leindler-type inequalities.

## Abstract

In this paper, we give a characterization of Nikol'ski\u{\i}-Besov type classes of functions, given by integral representations of moduli of smoothness, in terms of series over the moduli of smoothness. Also, necessary and sufficient conditions in terms of monotone or lacunary Fourier coefficients for a function to belong to a such a class are given. In order to prove our results, we make use of certain recent reverse Copson- and Leindler-type inequalities.