On the least deviations of the function $\operatorname{sign}x$ and its primitives from the rational functions in the $L_p$ metrics, $0<p\leqslant\infty$

In this paper estimates of weak equivalence type, as $n\to\infty$ are given for the least deviations $L_pR_n(f,[-1,1])$ of the functions $f(x)=x^s\operatorname{sign}x$ ($s=0,1,\dots$) in the metric of $L_p[-1,1]$ ($1\leqslant p\leqslant\infty$) from the rational functions of degree $\leqslant n$ ($n=1,2,\dots$). Specifically it is shown that
$$ L_pR_n(x^s\operatorname{sign}x,[-1,1])\asymp n^\frac1{2p}\exp\Biggl\{-\pi\sqrt{\biggl(s+\frac1p\biggr)n}\Biggr\} $$
($s\ne0$ при $p=\infty$); in particular,
\begin{gather*} L_pR_n(\operatorname{sign}x,[-1,1])\asymp n^\frac1{2p}\exp\Biggl\{-\pi\sqrt{\frac np}\Biggr\}\qquad(1\leqslant p<\infty), \\ L_pR_n(|x|,[-1,1])\asymp n^\frac1{2p}\exp\Biggl\{-\pi\sqrt{\biggl(1+\frac1p\biggr)n}\Biggr\}\qquad(1\leqslant p\leqslant\infty). \end{gather*}

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