Abstract:
In the Bergman space $B_{q,\gamma}$$(1\leq q<\infty$, $\gamma:=\gamma(|z|)> 0)$, we find sharp inequalities between the best simultaneous approximation of a function and the averaged moduli of smoothness $\omega_2(f^{(r)},t)_{H_{q,R}}$ of the angular boundary values of the $r$th derivatives. These inequalities are applied to the problem of evaluation of the supremum of best simultaneous approximations of some classes of functions defined in terms of moduli of smoothness and lying in the Bergman space $B_{q,\gamma}$.
Keywords:extremal problem, simultaneous approximation of functions and their derivatives, algebraic polynomial, modulus of continuity, Hardy space, Bergman space.
