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

Authors: Gensun Fang and Xuehua Li
Journal: Math. Comp. 75 (2006), 241-258
MSC (2000): Primary 65E05, 41A46; Secondary 30D55, 30E10
Published electronically: June 16, 2005
MathSciNet review: 2176398
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Abstract: 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 $L_{q}[0,2\pi]$, $r\in{\mathbb N}$, $1\leq q<\infty$.

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Additional Information

Gensun Fang
Affiliation: School of Mathematical Sciences, Beijing Normal University, Beijing 100875, People’s Republic of China

Xuehua Li
Affiliation: School of Mathematical Sciences, Beijing Normal University, Beijing 100875, People’s Republic of China

Keywords: Hardy--Sobolev class, $n$-widths, Kolmogorov type comparison theorem.
Received by editor(s): July 16, 2002
Received by editor(s) in revised form: January 8, 2004
Published electronically: June 16, 2005
Additional Notes: The authors were supported in part the Natural Science Foundation of China Grant #10371009 and Research Fund for the Doctoral Program Higher Education.
Article copyright: © Copyright 2005 American Mathematical Society

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