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

Estimates are obtained for the nonsymmetric deviations R_n sign x] and R_n sign x]_L of the function sign x from rational functions of degree ≤n, respectively, in the metric $$c( - 1, - \delta ] \cup \delta ,1]), 0< \delta< exp( - \alpha \surd \overline n ), \alpha > 0,$$ and in the metric L?1, 1]: $$\begin{gathered} R_n sign x] _{\frown }^\smile exp \{ - \pi ^2 n/(2 ln 1/\delta )\} , n \to \infty , \hfill \\ 10^{ - 3} n^{ - 2} \exp ( - 2\pi \surd \overline n )< R_n sign x_{|L}< \exp ( - \pi \surd \overline {n/2} + 150). \hfill \\ \end{gathered} $$ Let 0 < δ < 1, Δ (δ)=?1, ? δ] ∪ δ, 1]; $$\begin{gathered} R_n f;\Delta (\delta )] = R_n f] = inf max |f(x) - R(x)|, \hfill \\ R_n f; - 1,1] ]_L = R_n f]_L = \mathop {inf}\limits_{R(x)} \smallint _{ - 1}^1 |f(x) - R(x)|dx, \hfill \\ \end{gathered} $$ where R(x) is a rational function of order at most n. Bulanov 1] proved that for δ ε e^?n, e^?1] the inequality $$\exp \left( {\frac{{\pi ^2 n}}{{2\ln (1/\delta }}} \right) \leqslant R_n sign x] \leqslant 30 exp\left( {\frac{{\pi ^2 n}}{{2\ln (1/\delta + 4 ln ln (e/\delta ) + 4}}} \right)$$ is valid. The lower estimate in this inequality was previously obtained by Gonchar (2], cf. also 1]).