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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
Аннотация:
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 $L_{\bar{p}, \theta^{(2)}}^{*}(\mathbb{I}^{m})$.
Ключевые слова:
Lorentz space, Nikol'skii-Besov class, best approximation.
Образец цитирования:
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
Образцы ссылок на эту страницу:
https://www.mathnet.ru/rus/umj108 https://www.mathnet.ru/rus/umj/v6/i1/p16
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Страница аннотации: | 143 | PDF полного текста: | 48 | Список литературы: | 24 |
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