Abstract:
The paper studies extremal problems related to the best polynomial approximation of functions that are analytic in the unit disk and belong to the Bergman space $B_2$ with a finite norm $$\|f\|_{2}=\left(\frac{1}{\pi}\iint_{|z|<1}|f(z)|^{2}d\sigma\right)^{1/2}<\infty.$$ Let $$B^{(r)}_2:=\left\{f\in B_2:\|f^{(r)}\|_2<\infty\right\},\quad r\in\mathbb{Z_+}\quad (f^{(0)}\equiv f).$$ An exact theorem is proved between the value of the best approximation $E_{n-1}(f)_2$ and the value of the modulus of continuity of the mth order $\sin (\pi t/h)$$(0<h\le\pi/n)$ of functions $\omega_{m}(f^{(r)},t)_2$ averaged with the weight $f\in B^{(r)}_2.$ The connection between the proven theorem and the behavior of exact constants in the Jackson-Stechkin inequality for moduli of continuity $\omega_{m}(f^{(r)},t)_2$ is clarified. For the class of functions $W_{m}^{(r)}(\Phi)_2$, given a given monotonically increasing moharant $\Phi$, satisfying some restrictions, the exact values of various $n$-widths in $B_2$ space are calculated.
Keywords:modulus of continuity, Jackson – Stechkin inequalities, $n$-widths, Bergman space.
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