On widths of some classes of analytic functions in a circle

Mirgand Sh. Shabozov ; Muqim S. Saidusajnov

Ural mathematical journal, Tome 10 (2024) no. 2, pp. 121-130

Résumé

We calculate exact values of some $n$-widths of the class $W_{q}^{(r)}(\Phi),$ $r\in\mathbb{Z}_{+},$ in the Banach spaces $\mathscr{L}_{q,\gamma}$ and $B_{q,\gamma},$ $1\leq q\leq\infty,$ with a weight $\gamma$. These classes consist of functions $f$ analytic in the unit circle, their $r$th order derivatives $f^{(r)}$ belong to the Hardy space $H_{q},$ $1\leq q\leq\infty,$ and the averaged moduli of smoothness of boundary values of $f^{(r)}$ are bounded by a given majorant $\Phi$ at the system of points $\{\pi/(2k)\}_{k\in\mathbb{N}}$; more precisely, $$ \frac{k}{\pi-2}\int_{0}^{\pi/(2k)}\omega_{2}(f^{(r)},2t)_{H_{q,\rho}}dt\leq \Phi\left(\frac{\pi}{2k}\right) $$ for all $k\in\mathbb{N}$, $k>r.$

Détail

Keywords: Modulus of smoothness, The best approximation, $n$-widths, The best linear method of approximation