The best approximation and the values of the widths of some classes of analytical functions in the weighted Bergman space B_(2,γ)

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 $f^{\left(r\right)}$ in the weighted Bergman space $B_{2,\gamma }$. Also using the modulus of continuity of the $m$-th order of the derivative $f^{\left(r\right)}$, we introduce a class of functions $W_m^{\left(r\right)}(h,\Phi )$ 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 , 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 -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 $B_{2,\gamma }$.