Article
Ufa Mathematical Journal
Volume 15, Number 1, pp. 1-20
On estimates for orders of best $M$-term approximations
of multivariate functions in anisotropic Lorentz--Karamata spaces
Akishev G.A.
DOI:10.13108/2023-15-1-1
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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 $2\pi$-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.