Order of the trigonometric widths of the Nikol’skii-Besov classes with mixed metric in the metric of anisotropic Lorentz spaces

Abstract

In this paper we estimate the order of the triginometric width of the Nikol’skii–Besov classes B^ατ_p(Tⁿ) with mixed metric in the anisotropic Lorentz space L_qθ(Tⁿ) when 1<p= (p₁, . . . ,p_n) < 2 < q= (q₁, . . . ,q_n). The concept of a trigonometric width in the one-dimensional case was first introduce by R.S. Ismagilov and he established his estimates for certain classes in the space of continuous functions. For a function of several variables exact orders of trigonometric width of Sobolev class W^r_p, Nikol’skii class H^r_p in the space L_q are established by E.S. Belinsky, V.E. Majorov, Yu. Makovoz, G.G. Magaril-Ilyaev, V.N. Temlyakov. This problem for the Besov class B^r_pq was investigated by A.S. Romanyuk, D.B. Bazarkhanov. The trigonometric width for the anisotropic Nikol’skii-Besov classes B^ατ_pr(Tⁿ) in the metric of the anisotropic Lorentz spaces L_qθ(Tⁿ) was found by K.A. Bekmaganbetov and Ye. Toleugazy.