Estimates of the order of approximation of functions of several variables in the generalized Lorentz space

Gabdolla Akishev

doi:10.5644/SJM.22.01.08

Authors

Gabdolla Akishev Kazakhstan Branch, Lomonosov Moscow State University, Department of Fundamental, and Applied Mathematics, Astana, Republic Kazakhstan

DOI:

https://doi.org/10.5644/SJM.22.01.08

Keywords:

Lorentz space, Nikol'skii--Besov class, trigonometric polynomial, best approximation, hyperbolic cross

Abstract

In this paper, we consider an anisotropic symmetric space $X(\bar\varphi)$ of $2\pi$-periodic functions of $m$ variables, in particular, the generalized Lorentz space $L_{\bar{\psi},\bar{\tau}}^{*}(\mathbb{T}^{m})$ and the Nikol'skii--Besov class $S_{X(\bar{\varphi}),\bar{\theta}}^{\bar r}B$. We prove an embedding theorem for the Nikol'skii--Besov class into the generalized Lorentz space and establish an upper estimate for the best approximations of functions from the class $S_{X(\bar{\varphi}),\bar{\theta}}^{\bar r}B$ by trigonometric polynomials whose harmonic indices belong to the hyperbolic cross.

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