On $n$-widths, optimal quadrature formulas, and optimal recovery of functions analytic in a strip

Let $H_\infty(D_H)$ be the space of bounded analytic functions in the strip $D_H:=\{z\in\mathbf C:|\operatorname{Im} z|<H\}$. 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 $BX:=\{x\in X:\|x\|\leqslant1\}$. In this paper the exact values of the Kolmogorov $n$-widths $d_{2n}(B\widetilde H_\infty^{\mathbf R}(D_H), L_q[0,2\pi])$, 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 $BH_\infty(D_H)$.