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

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.