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(φ) of 2π-periodic functions of m variables, in particular, the generalized Lorentz space Lψ̄,τ̄*(Tᵐ) and the Nikol'skii--Besov class SX(φ̄),θ̄ʳB. 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 SX(φ̄),θ̄ʳB by trigonometric polynomials whose harmonic indices belong to the hyperbolic cross.