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On the best polynomial approximation of functions in the Hardy space $H_{q,R}, (1\le q\le\infty, R\ge 1)$

M. Sh. Shabozov^a, G. A. Yusupov<sup>b

a</sup> Tajik National University (Tajikistan, Dushanbe)
^b Tajik National University (Tajikistan, Dushanbe)

Abstract: Exact inequalities are found between the best polynomial approximation of functions analytics in the disk $U_R:=\bigl\{z\in\mathbb{C}, |z|<R\bigr\},$ $R\ge1$ and the averaged modulus of continuity angular boundary values of the $m$th order derivatives. For the class $W_{q,R}^{(m)} \ (m\in\mathbb{Z}_+,$ $1\le q\le\infty, R\ge1)$ of functions $f\in H_{q,R}^{(m)}$ whose $m$-order derivatives $f^{(m)}$ belong to the Hardy space $H_{q,R}$ and satisfy the condition $\|f ^{(m)}\|_{q,R}\le1,$ the exact values of the upper bounds of the best approximations are calculated. Moreover, for the class $W^{(m)}_{q,R}(\Phi),$ consisting of all functions $f\in H_{q,R}^{(m)},$ for which any $k\in\mathbb{N}, m\in\mathbb{Z}_{+}, k>m$ the averaged moduli of continuity of the boundary values of the $m$th order derivative $f^{(m )},$ dominated in the system of points $\{\pi/k\}_{k\in\mathbb{N}}$ by the given function $\Phi,$ satisfy the condition
\begin{equation*} \int\limits_{0}^{\pi/k}\omega\bigl(f^{(m)},t\bigr)_{q,R}dt\le\Phi(\pi/k), \end{equation*}
the exact values of the Kolmogorov and Bernstein $n$-widths are calculated in the norm of the space $H_{q} \ (1\le q\le\infty).$
The results obtained generalize some results of L.V.Taikov on classes of analytic functions in a circle of radius $R\ge1.$

Keywords: the best approximation, Hardy space, modulus of continuity, majorizing function, $n$-widths.

UDC: 517.5

Received: 23.11.2022
Accepted: 24.04.2023

DOI: 10.22405/2226-8383-2023-24-1-182-193