ON WIDTHS OF SOME CLASSES OF ANALYTIC FUNCTIONS IN A CIRCLE

We calculate exact values of some $n$-widths of the class \(W_{q}^{(r)}(\Phi),\) \(r\in\mathbb{Z}_{+},\) in the Banach spaces \(\mathscr{L}_{q,\gamma}\) and \(B_{q,\gamma},\) \(1\leq q\leq\infty,\) with a weight \(\gamma\). These classes consist of functions \(f\) analytic in the unit circle, their \(r\)th order derivatives \(f^{(r)}\) belong to the Hardy space \(H_{q},\) \(1\leq q\leq\infty,\) and the averaged moduli of smoothness of boundary values of \(f^{(r)}\) are bounded by a given majorant \(\Phi\) at the system of points \(\{\pi/(2k)\}_{k\in\mathbb{N}}\); more precisely,$$ \frac{k}{\pi-2}\int_{0}^{\pi/(2k)}\omega_{2}(f^{(r)},2t)_{H_{q,\rho}}dt\leq \Phi\left(\frac{\pi}{2k}\right) $$for all \(k\in\mathbb{N}\), \(k>r.\)