Approximation of the function sign x in the uniform and integral metrics by means of rational functions

S. A. Agahanov   ; N. Sh. Zagirov

Matematičeskie zametki, Tome 23 (1978) no. 6, pp. 825-838

Résumé

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]).