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This article is cited in 1 scientific paper (total in 1 paper)
Approximation of the function sign x in the uniform and integral metrics by means of rational functions
S. A. Agahanov, N. Sh. Zagirov Daghestan State University
Abstract:
Estimates are obtained for the nonsymmetric deviations $R_n[\operatorname{sign}x]$ and $R_n[\operatorname{sign}x]_L$ of the function $\operatorname{sign}x$ from rational functions of degree $\le n$, respectively, in the metric
$$
C([-1,-\delta]\cup[\delta,1]),\quad0<\delta<\exp(-\alpha\sqrt{n}),\quad\alpha>0,
$$
and in the metric $L[-1,1]$:
\begin{gather*}
R_n[\operatorname{sign}x]\asymp\exp\{-\pi^2n/(2\ln1/\delta)\},\quad n\to\infty,\\
10^{-3}n^{-3}\exp(-2\pi\sqrt{n})<R_n[\operatorname{sign}x]_L<\exp(-\pi\sqrt{n/2}+150).
\end{gather*}
is valid. The lower estimate in this inequality was previously obtained by Gonchar ([2], cf. also [1]).
Received: 29.04.1976
Citation:
S. A. Agahanov, N. Sh. Zagirov, “Approximation of the function sign x in the uniform and integral metrics by means of rational functions”, Mat. Zametki, 23:6 (1978), 825–838; Math. Notes, 23:6 (1978), 452–460
Linking options:
https://www.mathnet.ru/eng/mzm8183 https://www.mathnet.ru/eng/mzm/v23/i6/p825
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Abstract page: | 208 | Full-text PDF : | 94 | First page: | 2 |
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