Abstract
We consider Hardy spaces \(H_{q,\rho }\) with \(1\le q\le \infty\) and \(0<\rho \le R\) and weighted Bergman spaces \(\mathscr {B}_{q,\gamma }\) and \(\mathscr {L}_{q,\gamma }\) with \(1\le q<\infty\) and \(\gamma \ge 0\) . We introduce classes \(W_{a}^{(r)}H_{q,R}(\Phi )\) of functions \(f\in H_{q,R}\) such that the \(r\) th derivative \(f_{a}^{(r)}\) with respect to the argument \(t\) of the complex variable \(z=\rho \exp (it)\) belongs to \(H_{q,R}\) and the condition \( \frac {1}{h}\int _{0}^{h}\omega(f_{a}^{(r)},t)_{H_{q,R}}dt\le \Phi (h)\) holds,where \(h\in\mathbb {R}_+\) and \(\omega (\varphi ,\cdot)_{H_{q,R}}\) denotes the modulus of continuity ofa function \(\varphi \) in \(H_{q,R}\) . We find the best linear approximation methods for \(W_{a}^{(r)}H_{q,R}(\Phi )\) and the exact valuesof a series of \(n\) -widths of this class in the above-mentioned spaces.
