On exact estimates of the order of approximation of functions of several variables in the anisotropic Lorentz-Zygmund space

TL;DR

This paper derives precise estimates for how well functions in a specific anisotropic Lorentz-Zygmund space can be approximated by trigonometric polynomials from a Nikol'skii-Besov class, highlighting the approximation order's exactness.

Contribution

It provides the first order-sharp approximation estimates in anisotropic Lorentz-Zygmund spaces for functions from Nikol'skii-Besov classes using hyperbolic cross polynomials.

Findings

Established order-sharp approximation estimates

Derived bounds for approximation by hyperbolic cross polynomials

Focused on anisotropic Lorentz-Zygmund spaces

Abstract

In this paper we consider $L_{\overline{p}, \overline\alpha, \overline{\tau}}^{*}(\mathbb{T}^{m})$ anisotropic Lorentz-Zyg\-mu\-nd space $ 2\pi$ of periodic functions of $m$ variables and Nikol'skii--Besov's class $S_{\overline{p}, \overline\alpha, \overline{\tau}, \bar{\theta}}^{\bar r}B$. In this paper, we establish order-sharp estimates of the best approximation by trigonometric polynomials with harmonic numbers from the step hyperbolic cross of functions from the Nikol'skii - Besov class in the norm of the anisotropic Lorentz-Zygmund space.