On the best polynomial approximation of functions in the weight Bergman space

The problem of finding an accurate estimate of the best approximation value $E_{n-1}(f)_{p},$ $1\leq p\leq\infty,$ using the average value of the modulus of continuity and the modulus of smoothness of the function and its corresponding derivatives is one of the important and interesting problems in the approximation theory. N. P. Korneychuk considered this problem for classes of $2\pi$ periodic functions with a convex modulus of continuity in the metric space of continuous functions $C[0, 2\pi].$ A similar problem without assuming convexity of the modulus of continuity was considered L. V. Taikov in the Hardy space $H_{p},$ $1\leq p\leq\infty$. Continuing this study of the Hardy spaces $H_{p},$ $p\geq 1,$ M. Sh. Shabozov and M. M. Mirkalonova proved new sharp inequalities in which the best approximation of analytic functions is estimated by the sums of averaged values of the modules of continuity of the function and some of its derivatives. In this paper, we give some sharp inequalities between the best polynomial approximations of analytic in the unit disk functions by algebraic complex polynomials and moduli of continuity and smoothness of a function itself and its second derivative in weighted Bergman spaces. The exact values of Bernstein and Kolmogorov $n$-widths of classes of functions in weighted Bergman spaces are calculated. The last theorem of this work generalizes a result by L. V. Taikov obtained for classes of differentiable periodic functions, to the case of functions analytic in the unit circle belonging to the space $B_{q,\gamma}$, $1\leq q\leq\infty$.