The Exact Values of Widths of Classes of Analytic in a DiscFunctions and the Best Linear Approximation Methods in Hardy and BergmanSpaces

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 of a function $$ \varphi $$ in $$H_{q,R} $$ . We find the best linear approximation methods for $$ W_{a}^{(r)}H_{q,R}(\Phi )$$ and the exact values of a series of $$n $$ -widths of this class in the above-mentioned spaces.