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

Abstract

Let $A(U)$ be a set of functions analytic in the circle $U:=\{z\in\mathds{C},\, |z|<1\}$ 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 $B_2$.