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In this paper we consider anisotropic Lorentz-Karamata space $2\pi$ ofperiodic functions of $m$ variables and Nikol'skii--Besov's class . In thispaper, we establish order-sharp estimates of the best approximation bytrigonometric polynomials with harmonic numbers from the step hyperbolic crossof functions from the Nikol'skii - Besov class in the norm of the anisotropicLorentz-Karamata space.

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