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

Estimates are obtained for the nonsymmetric deviations Rn [sign x] and Rn [sign x]L of the function sign x from rational functions of degree ≤n, respectively, in the metric $$c([ - 1, - \delta ] \cup [\delta ,1]), 0 0,$$ 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.