On best approximation of functions in Bergman space $B_2$.

Abstract

In the paper we study extremal problems related to the best polynomial approximation of functions analytic in the unit disk and belonging to the Hilbert Bergman space $B_2$. We find exact inequalities for the best approximation of an arbitrary function $f\in B_2$, analytic in the unit disk, by algebraic complex polynomials $p_n\in \mathcal{P}_n$ by means of the averaged value of the modulus of continuity $\omega(f^{(r)},t)_{B_2}$ of the $r$th derivative $f^{(r)}$ in
the space $B_2$. We introduce the class $W^{(r)}_2(\omega,\Phi)$ of functions analytic in the unit disk whose averaged value of the modulus of continuity of the derivative $f^{(r)}$ satisfies the inequality
$$
\int\limits_{0}^{u}\omega^2(f^{(r)},t)_{B_2}\sin\frac{\pi}{u}t d t\leq
\Phi^2(u), \qquad 0\leq u\leq 2\pi.
$$
For certain restrictions for majorant $\Phi$, we calculate exact values of various $n$-widths for the introduced class of functions. To solve the mentioned problems, we use the methods of solving extremal problems in normed spaces and we use the method for estimating $n$-widths developed by V.M.Tikhomirov.