On the best approximation and the values of the widths of some classes of functions in the Bergmann weight space

We consider the extremal problem of finding exact constants in the Jackson–Stechkin type inequalities connecting the best approximations of analytic in the unit circle $U=\{z:|z|<1\}$ functions by algebraic complex polynomials and the averaged values of the higher-order continuity modules of the $r$-th derivatives of functions in the Bergman weight space $B_{2,\gamma}.$ The classes of analytic in the unit circle functions $W_{m}^{(r)}(\tau)$ and $W_{m}^{(r)}(\tau,\Phi)$ which satisfy some specific conditions are introduced. For the introduced classes of functions, the exact values of some known $n$-widths are calculated. In this paper, we use the methods of solving extremal problems in normalized spaces of functions analytic in a circle and a well-known method developed by V. M. Tikhomirov for estimating from below the $n$-widths of functional classes in various Banach spaces. The results obtained in the work generalize and extend the results of the works by S. B. Vakarchuk and A. N. Shchitova obtained for the classes of differentiable periodic functions to the case of analytic in the unit circle functions belonging to the Bergmann weight space.