Estimates of the order of approximation of functions of several variables in the generalized Lorentz space

TL;DR

This paper investigates the approximation properties of functions in anisotropic Lorentz and Nikol'skii--Besov spaces, establishing embedding theorems and bounds for best trigonometric polynomial approximations in multiple variables.

Contribution

It introduces new embedding results and approximation bounds for functions in generalized Lorentz and Nikol'skii--Besov spaces of several variables.

Findings

Proved an embedding theorem for Nikol'skii--Besov and Lorentz spaces.

Established upper bounds for best trigonometric polynomial approximations.

Analyzed approximation rates using hyperbolic cross methods.

Abstract

In this paper we consider $ X(\bar\varphi)$ anisotropic symmetric space $ 2\pi$ of periodic functions of $m$ variables, in particular, the generalized Lorentz space $L_{\bar{\psi},\bar{\tau}}^{*}(\mathbb{T}^{m})$ and Nikol'skii--Besov's class $S_{X(\bar{\varphi}),\bar{\theta}}^{\bar r}B$. The article proves an embedding theorem for the Nikol'skii - Besov class in the generalized Lorentz space and establishes an upper bound for the best approximations by trigonometric polynomials with harmonic numbers from the hyperbolic cross of functions from the class $S_{X(\bar{\varphi}),\bar{\theta}}^{\bar r}B$.