The best approximation of analytic in a unit circle functionsin the Bergman weight space $\mathscr{B}_{2,\mu}$

The paper studies the issues of the best approximation of analytical functions in the Bergman weight space $\mathscr{B}_{2,\mu}.$ In this space, for best approximations of functions analytic in the circle by algebraic complex polynomials we obtain the exact inequalities by means of generalized modules of continuity of higher order derivatives $\Omega_{m}(z^{r}f^{(r)},t),$ $m\in\mathbb{N},$ $r\in\mathbb{Z}.$ For classes of functions analytic in the unit circle defined by the characteristic $\Omega_{m}(z^{r}f^{(r)},t),$ and the majorant $\Phi,$ the exact values of some $n$-widths are calculated. When proving the main results of this work, we use methods for solving extremal problems in normalized spaces of functions analytic in the circle, N. P. Korneichuk’s method for estimating upper bounds for the best approximations of classes of functions by a subspace of fixed dimension, and a method for estimating from below the $n$-widths of function classes in various Banach spaces.