ON $n$-WIDTHS, OPTIMAL QUADRATURE FORMULAS, AND OPTIMAL RECOVERY OF FUNCTIONS ANALYTIC IN A STRIPстатья
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Аннотация:Let $H_\infty(D_H)$ be the space of bounded analytic functions in the strip $D_H:=\{z\in\mathbf{C}\colon\vert\operatorname{Im}z\vert< H\}$. We denote by $\widetilde{H}_\infty(D_H)$ the set of $2\pi$-periodic functions in $H_\infty(D_H)$, and by $\widetilde{H}_\infty^{\mathbf{R}}(D_H)$ the set of functions in $\widetilde{H}_\infty(D_H)$ that are real on the real axis. For a normed linear space $X$ we set $BX:=\{x\in X\colon\Vert x\Vert \leqslant 1\}$. In this paper the exact values of the Kolmogorov $n$-widths $d_{2n}(B\widetilde{H}_\infty^{\mathbf{R}}(D_H)$, $L_q\lbrack 0, 2\pi\rbrack$ are found for all $1 \leqslant q\leqslant\infty$, an optimal quadrature formula is constructed for the class $B\widetilde{H}_\infty(D_H)$ by using the values of functions defined with an error and it is proved that the unique (to within a shift) optimal system of nodes is given by a uniform net. In addition to this, a number of problems are solved for the optimal recovery of functions and their derivatives in the class $BH_\infty(D_H)$.