On orders of approximation functions of generalized smoothness in Lorentz spaces

TL;DR

This paper investigates how well generalized Nikol'skii-Besov classes of periodic functions of multiple variables can be approximated by Fourier series in Lorentz spaces with mixed norms, providing estimates for the approximation order.

Contribution

It offers new estimates for the approximation order of generalized Nikol'skii-Besov classes in Lorentz spaces with mixed norms, extending previous results to multivariate periodic functions.

Findings

Derived bounds for approximation errors by Fourier partial sums

Extended approximation theory to Lorentz spaces with mixed norms

Provided new insights into the approximation of multivariate periodic functions

Abstract

This paper considers the Lorentz space with mixed norm of periodic functions of many variables and of the generalized Nikol'skii -- Besov classes. Estimates for the order of approximation of the generalized Nikol'skii -- Besov classes by partial sums of Fourier's series for multiple trigonometric system in Lorentz spaces with mixed norm are obtained.