Abstract:
In Hardy spaces $H_{q,\rho} (1\le q\le\infty,\ 0<\rho\le R)$ and weighted Bergman spaces $\mathscr{B}_{q,\gamma}$ and $\mathscr{L}_{q,\gamma}$$(1\le q<\infty, \gamma\ge0)$, the best linear approximation methods for classes $W_{a}^{(r)}H_{q,R}(\Phi)$ of functions are obtained. These are functions $f\in H_{q,R}$ whose $r$-th derivative $f_{a}^{(r)}$ with respect to the argument $t$ of the complex variable $z=\rho\exp(it)$ also belongs to $H_{q,R}$ and satisfies the condition $$\frac{1}{h}\int_{0}^{h}\omega(f_{a}^{(r)},t)_{H_{q,R}} dt\le\Phi(h),$$ where $h\in\mathbb{R}_+$, $\omega(\varphi,t)_{H_{q,R}}$ is the modulus of continuity of the function $\varphi\in H_{q,R}$. The exact values of certain $n$-widths of the class $W_{a}^{(r)}H_{q,R}(\Phi)$ in the mentioned spaces are calculated.
Key words:best linear approximation methods, modulus of continuity, Hardy space, weighted Bergman space, $n$-widths.
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