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

Abstract

In this paper we consider $X(\bar{\phi })$ anisotropic symmetric space $2\pi$ of periodic functions of $m$ variables, in particular, the generalized Lorentz space $L_{\bar{\psi },\bar{\tau }}^{\ast }({\mathbb{T}}^m)$ and Nikol'skii--Besov's class $S_{X(\bar{\phi }),\bar{\theta }}^{\bar{r}}B$. 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 $S_{X(\bar{\phi }),\bar{\theta }}^{\bar{r}}B$.