Abstract
In the Banach spaces \({\boldsymbol{\mathcal{L}}}_{q,\gamma}\) and \(B_{q,\gamma}\) \((1\leq q\leq\infty)\) with weight \(\gamma\) , the sharp values of some \(n\) -widths of the classes \(W_{a}^{(r)}H_{q}(\Phi,\mu)\) , \(r\in\mathbb{N}\) , \(1\leq q\leq\infty\) , \(\mu\in\mathbb{R}\) , \(\mu\geq 1\) are evaluated. These classes consist of functions \(f\in H_{q,a}^{(r)}\) such that the averaged modulus of continuity of their derivatives \(f_{a}^{(r)}\) are upper bounded by a given majorant \(\Phi\) . We also specify a best linear method on which the sharp value of the linear \(n\) -width for the classes \(W_{a}^{(r)}H_{q}(\Phi,\mu)\) in \({\boldsymbol{\mathcal{L}}}_{q,\gamma}\) is realized.
