On mean–square approximation of functions in Bergman space $B_2$ and value of widths of some classes of functions

Let $A(U)$ be a set of functions analytic in the circle $U:=\{z\in\mathbb{C}, |z|<1\}$ and $B_{2}:=B_{2}(U)$ be the space of the functions $f\in A(U)$ with a finite norm
$$\|f\|_{2}=\left(\frac{1}{\pi}\iint_{(U)}|f(z)|^{2} d\sigma\right)^{\frac{1}{2}}<\infty,$$
where $d\sigma$ is the area differential and the integral is treated in the Lebesgue sense. In the work we study extremal problems related with the best polynomial approximation of the functions $f\in A(U)$. We obtain a series of sharp theorems and calculate the values of various $n$–widths of some classes of functions defined by the continuity moduluses of $m$th order for the $r$th derivative $f^{(r)}$ in the space $B_2$.