Comparison theorems of Kolmogorov type and exact values of $n$-widths on Hardy-Sobolev classes

Let $S_{\beta}:=\{z\in {\mathbb C}:\vert{\rm Im}z\vert<\beta\}$ be a strip in complex plane. $\widetilde{H}_{\infty,\beta}^{r}$ denotes those $2\pi$-periodic, real-valued functions on ${\mathbb R}$ which are analytic in the strip $S_{\beta}$ and satisfy the condition $\vert f^{(r)}(z)\vert\leq 1$, $z\in S_{\beta}$. Osipenko and Wilderotter obtained the exact values of the Kolmogorov, linear, Gel'fand, and information $n$-widths of $\widetilde{H}_{\infty,\beta}^{r}$ in $L_{\infty}[0,2\pi]$, $r=0,1,2,\ldots$, and 2$n$-widths of $\widetilde{H}_{\infty,\beta}^{r}$ in $L_{q}[0,2\pi]$, $r=0$, $1\leq q<\infty$.

In this paper we continue their work. Firstly, we establish a comparison theorem of Kolmogorov type on $\widetilde{H}_{\infty,\beta}^{r}$, from which we get an inequality of Landau-Kolmogorov type. Secondly, we apply these results to determine the exact values of the Gel'fand $n$-width of $\widetilde{H}_{\infty,\beta}^{r}$ in $L_{q}[0,2\pi]$, $r=0,1,2\ldots,$ $1\leq q<\infty$. Finally, we calculate the exact values of Kolmogorov $2n$-width, linear $2n$-width, and information $2n$-width of $\widetilde{H}_{\infty,\beta}^{r}$ in $L_{q}[0,2\pi]$, $r\in{\mathbb N}$, $1\leq q<\infty$.