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 ; G. A. Yusupov

Čebyševskij sbornik, Tome 24 (2023) no. 1, pp. 182-193

Résumé

Exact inequalities are found between the best polynomial approximation of functions analytics in the disk $U_R:=\bigl\{z\in\mathbb{C}, |z|$ $R\ge1$ 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 $R\ge1.$

Détail

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