Abstract
In the paper we consider a well-known class of weakly varying functions and by these functions we define an anisotropic Lorentz-Karamata space of 2𝜋-periodic functions of many variables.Particular cases of these spaces are anisotropic Lorentz-Zygmund and Lorentz spaces.In the anisotropic Lorentz-Karamata space we define an analogue of Nikolskii-Besov space.The main aim of the paper is to find sharp orders of best 𝑀 -term trigonometric approximation of functions from Nikolskii-Besov space by the norm of another anisotropic Lorentz-Karamata space.In the paper we establish order sharp two-sided estimates of best 𝑀 -term trigonometric approximations for the functions from the Nikolskii-Besov space in the anisotropic Lorentz-Karamata space in various metrics.In order to prove an upper bound for 𝑀 -term approximations, we employ an idea of the greedy algorithms proposed by V.N.Temlyakov and we modify it for the anisotropic Lorentz-Karamata space.