On the Best Simultaneous Approximation in the Bergman Space $B_2$

We study extremal problems related to the best joint polynomial approximation of functions analytic in the unit disk and belonging to the Bergman space $B_{2}$. The problem of joint approximation of periodic functions and their derivatives by trigonometric polynomials was considered by Garkavy [1:x435] in 1960. Then, in the same year, Timan [2:x435] considered this problem for classes of entire functions defined on the entire axis. The problem of joint approximation of functions and their derivatives is considered in more detail in Malozemov's monograph [3:x435], where some classical theorems of the theory of approximation of functions are presented and generalized. In the present paper, a number of exact theorems are obtained and sharp upper bounds for the best joint approximations of a function and its successive derivatives by polynomials and their respective derivatives on some classes of complex functions belonging to the Bergman space $B_{2}$ are calculated.