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

In this paper, we consider an anisotropic symmetric space $X(\bar\varphi)$ of $2\pi$-periodic functions of $m$ variables, in particular, the generalized Lorentz space $L_{\bar{\psi},\bar{\tau}}^{*}(\mathbb{T}^{m})$ and the Nikol'skii--Besov class $S_{X(\bar{\varphi}),\bar{\theta}}^{\bar r}B$. We prove an embedding theorem for the Nikol'skii--Besov class into the generalized Lorentz space and establish an upper estimate for the best approximations of functions from the class $S_{X(\bar{\varphi}),\bar{\theta}}^{\bar r}B$ by trigonometric polynomials whose harmonic indices belong to the hyperbolic cross.