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

Abstract:

Let $S_{\beta }:=\{z\in {\mathbb C}:|\textrm {Im}z|<\beta \}$ 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 $|f^{(r)}(z)|\leq 1$, $z\in S_{\beta }$. 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 $L_{q}[0,2\pi ]$, $r\in {\mathbb N}$, $1\leq q<\infty$.