Jackson–Stechkin type inequalities and widths of classes of functions in the weighted Bergman space

In extremal problems of the theory of approximation of functions an important role is played be exact inequalities of the value of the best polynomial approximation by means of averaged values of the modules of continuity of higher orders of the derived functions. In this paper we present an inequality of type Ligun-two-sided estimate for the best weighted approximate analytic functions in the unit disc from the Bergman space $B_{2,\gamma}.$ The resulting inequalities allow us to establish new connections between the constructive and structural properties of the functions and for the corresponding classes of functions give an estimate from the top of the widths. The exact values of Bernstein, Kolmogorov, Gelfand, linear and projection $n$-widths of classes of analytic functions in unit discs defined by modules of continuity of higher orders of the derived functions in the space $B_{2,\gamma}$ averaged with positive weight are calculated.