Abstract
In this paper, a number of extreme problems related to the best polynomial approximation of functions analytical in the disk \(U: = \left\{ {z \in \mathbb{C}:\left| z \right| < 1} \right\}\) and belonging to the Bergman space \({{B}_{2}}\) are solved. The bilateral inequality is proved, which is a generalization of the result for periodic functions \(f \in {{L}_{2}}\) obtained by Shabozov and Yusupov for the class \(L_{2}^{{(r)}}[0,2\pi ]\) , in which the derivative \({{f}^{{(r - 1)}}}\) is absolutely continuous and the derivative of the \(r\) th order \({{f}^{{(r)}}} \in {{L}_{2}}\) in the case of a polynomial approximation of \(f \in \mathcal{A}(U)\) belongs to \(B_{2}^{{(r)}}(U)\) . A number of cases are given when the bilateral inequality turns into equality. For some classes of functions belonging to \({{B}_{2}}\) , the exact values of the known \(n\) -widths are found, and the problem of joint approximation of functions and their intermediate derivatives is solved.
