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Ural Math. J., 2020, Volume 6, Issue 1, Pages 16–29 (Mi umj108)  

Estimates of best approximations of functions with logarithmic smoothness in the Lorentz space with anisotropic norm

Gabdolla Akishevab

a L.N. Gumilyov Eurasian National University
b Ural Federal University named after the First President of Russia B. N. Yeltsin, Ekaterinburg

Abstract: In this paper, we consider the anisotropic Lorentz space $L_{\bar{p}, \bar\theta}^{*}(\mathbb{I}^{m})$ of periodic functions of $m$ variables. The Besov space $B_{\bar{p}, \bar\theta}^{(0, \alpha, \tau)}$ of functions with logarithmic smoothness is defined. The aim of the paper is to find an exact order of the best approximation of functions from the class $B_{\bar{p}, \bar\theta}^{(0, \alpha, \tau)}$ by trigonometric polynomials under different relations between the parameters $\bar{p}, \bar\theta,$ and $\tau$.
The paper consists of an introduction and two sections. In the first section, we establish a sufficient condition for a function $f\in L_{\bar{p}, \bar\theta^{(1)}}^{*}(\mathbb{I}^{m})$ to belong to the space $L_{\bar{p}, \theta^{(2)}}^{*}(\mathbb{I}^{m})$ in the case $1{<\theta^{2}<\theta_{j}^{(1)}},$ ${j=1,\ldots,m},$ in terms of the best approximation and prove its unimprovability on the class $E_{\bar{p},\bar{\theta}}^{\lambda}=\{f\in L_{\bar{p},\bar{\theta}}^{*}(\mathbb{I}^{m})\colon {E_{n}(f)_{\bar{p},\bar{\theta}}\leq\lambda_{n},}$ ${n=0,1,\ldots\},}$ where $E_{n}(f)_{\bar{p},\bar{\theta}}$ is the best approximation of the function $f \in L_{\bar{p},\bar{\theta}}^{*}(\mathbb{I}^{m})$ by trigonometric polynomials of order $n$ in each variable $x_{j},$ $j=1,\ldots,m,$ and $\lambda=\{\lambda_{n}\}$ is a sequence of positive numbers $\lambda_{n}\downarrow0$ as $n\to+\infty$. In the second section, we establish order-exact estimates for the best approximation of functions from the class $B_{\bar{p}, \bar\theta^{(1)}}^{(0, \alpha, \tau)}$ in the space $L_{\bar{p}, \theta^{(2)}}^{*}(\mathbb{I}^{m})$.

Keywords: Lorentz space, Nikol'skii-Besov class, best approximation.

Funding Agency Grant Number
Ministry of Education and Science of the Russian Federation 02.A03.21.0006
This work was supported by the Competitiveness Enhancement Program of the Ural Federal University (Enactment of the Government of the Russian Federation of March 16, 2013 no. 211, agreement no. 02.A03. 21.0006 of August 27, 2013).


DOI: https://doi.org/10.15826/umj.2020.1.002

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Citation: Gabdolla Akishev, “Estimates of best approximations of functions with logarithmic smoothness in the Lorentz space with anisotropic norm”, Ural Math. J., 6:1 (2020), 16–29

Citation in format AMSBIB
\Bibitem{Aki20}
\by Gabdolla Akishev
\paper Estimates of best approximations of functions with logarithmic smoothness in the Lorentz space with anisotropic norm
\jour Ural Math. J.
\yr 2020
\vol 6
\issue 1
\pages 16--29
\mathnet{http://mi.mathnet.ru/umj108}
\crossref{https://doi.org/10.15826/umj.2020.1.002}
\mathscinet{http://www.ams.org/mathscinet-getitem?mr=MR4128757}
\zmath{https://zbmath.org/?q=an:07255684}
\elib{https://elibrary.ru/item.asp?id=43793621 }
\scopus{https://www.scopus.com/record/display.url?origin=inward&eid=2-s2.0-85089116548}


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