Geodesic
Inequalities for the best ``angular'' approximation and the smoothness modulus of a function in the Lorentz space
Itogi nauki i tehniki. Sovremennaâ matematika i eë priloženiâ. Tematičeskie obzory, Proceedings of the Voronezh international spring mathematical school "Modern methods of the theory of boundary-value problems. Pontryagin readings—XXXIV", Voronezh, May 3-9, 2023, Part 1, Tome 230 (2023), pp. 8-24

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In this paper, we consider the Lorentz space $L_{p, \tau}(\mathbb{T}^{m})$ of $2\pi$-periodic functions of several variables, the best “angular” approximation of such functions by trigonometric polynomials, and the mixed smoothness modulus of functions from this space. The properties of the mixed smoothness modulus are given and strengthened versions of the direct and inverse theorems on the “angular” approximations are proved.
Keywords: Lorentz space, trigonometric polynomial, best “angular” approximation, smoothness modulus 
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     author = {G. A. Akishev},
     title = {Inequalities for the best ``angular'' approximation and the smoothness modulus of a function in the {Lorentz} space},
     journal = {Itogi nauki i tehniki. Sovremenna\^a matematika i e\"e prilo\v{z}eni\^a. Temati\v{c}eskie obzory},
     pages = {8--24},
     publisher = {mathdoc},
     volume = {230},
     year = {2023},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/INTO_2023_230_a1/}
}
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G. A. Akishev. Inequalities for the best ``angular'' approximation and the smoothness modulus of a function in the Lorentz space. Itogi nauki i tehniki. Sovremennaâ matematika i eë priloženiâ. Tematičeskie obzory, Proceedings of the Voronezh international spring mathematical school "Modern methods of the theory of boundary-value problems. Pontryagin readings—XXXIV", Voronezh, May 3-9, 2023, Part 1, Tome 230 (2023), pp. 8-24. http://geodesic.mathdoc.fr/item/INTO_2023_230_a1/