The best approximation and the values of the widths of some classes of analytical functions in the weighted Bergman space $\mathscr{B}_{2,\gamma}$

In the paper, exact inequalities are found for the best approximation of an arbitrary analytic function $f$ in the unit circle by algebraic complex polynomials in terms of the modulus of continuity of the $m$th order of the $r$th order derivative $f^{(r)}$ in the weighted Bergman space $ \mathscr{B}_{2,\gamma}.$ Also using the modulus of continuity of the $m$-th order of the derivative $f^{(r)}$, we introduce a class of functions $W_{m}^{(r)}(h,\Phi)$ analytic in the unit circle and defined by a given majorant $\Phi,$ $h\in (0,\pi/n],$ $n>r,$ monotonically increasing on the positive semiaxis. Under certain conditions on the majorant $\Phi,$ for the introduced class of functions, the exact values of some known $n$-widths are calculated. We use methods for solving extremal problems in normed spaces of functions analytic in a circle, as well as the method for estimating from below the $n$-widths of functional classes in various Banach spaces developed by V. M. Tikhomirov. The results presented in this paper are a continuation and generalization of some earlier results on the best approximations and values of widths in the weighted Bergman space $\mathscr{B}_{2,\gamma}.$