Geodesic
On estimates for orders of best $M$-term approximations
Ufa mathematical journal, Tome 15 (2023) no. 1, pp. 1-20

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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.
Keywords: $M$–term approximation.
Mots-clés : Lorentz-Karamata space, Nikolskii-Besov space 
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G. A. Akishev. On estimates for orders of best $M$-term  approximations. Ufa mathematical journal, Tome 15 (2023) no. 1, pp. 1-20. http://geodesic.mathdoc.fr/item/UFA_2023_15_1_a0/