On the Best Polynomial Approximation of Analytical Functions in the Bergman Space B2

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.