The best joint approximation of some classes of functions in the Bergman space $B_{2}$

In this paper, we study several extreme problems related to the best joint approximation of certain classes of analytical functions in the unit circle given by higher-order continuity modules in the Bergman space $B_2$. It should be noted that for the first time the problem of joint approximation of periodic differentiable functions and their consecutive derivatives by trigonometric polynomials and their corresponding derivatives in a uniform metric was investigated by A.L.Garkavi [1]. The results obtained in [1] were generalized by A.F.Timan [2] for a class of integer functions of exponential type on the entire line. In the monograph [3]. The problems of joint approximation are generalized to some classical theorems of the theory of approximation of functions. However, in the listed works, only asymptotically accurate results were obtained. In this paper, we prove a number of exact theorems for the joint approximation of analytic functions in the unit circle belonging to the Bergman space $B_2$, complementing the results of M.Sh.Shabozov [4].