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 p (Tn) with mixed metric in the anisotropic Lorentz space Lq (Tn) when 1<p= (p1; : : : ;pn) < 2 < q= (q1; : : : ;qn). 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 Wr p , Nikol’skii class Hr p in the space Lq are established by E.S. Belinsky, V.E. Majorov, Yu. Makovoz, G.G. Magaril-Ilyaev, V.N. Temlyakov. This problem for the Besov class Br pq was investigated by A.S. Romanyuk, D.B. Bazarkhanov. The trigonometric width for the anisotropic Nikol’skii-Besov classes B pr (Tn) in the metric of the anisotropic Lorentz spaces Lq (Tn) was found by K.A. Bekmaganbetov and Ye. Toleugazy.