$p(x)/q(x)$ of degree $d$ gives rise to a degree-$2d$ rational approximant with the same error and a positive denominator, namely, $\{p(x)q(x)\}/q(x)^2.$

The infimum in the definitions of $R(f,d)$ and $R^+(f,d)$ cannot in general be replaced by a minimum [@rivlin-book], even when $X$ is finite subset of $\Re.$ This contrasts with the more familiar setting of a finite-dimensional normed linear space, where least-error approximants are guaranteed to exist.

For $S\subseteq\Re,$ we let $$\begin{aligned}
 R^+(S,d) = \inf_{\rule{0pt}{7pt}p,q}\, \sup_{x\in S} 
   \left\lvert \sign x - \frac{p(x)}{q(x)} \right\rvert,\end{aligned}$$ where the infimum ranges over $p,q\in P_d$ such that $q$ is positive on $S.$ The study of the rational approximation of the sign function dates back to seminal work by Zolotarev [@zolotarev1877rational] in the late 19th century. A much later result due to Newman [@newman64rational] gives highly accurate estimates of $R^+([-n,-1]\cup[1,n],d)$ for all $n$ and $d.$ Newman’s work in particular provides upper bounds on $R^+(\{\pm1,\pm2,\dots,\pm n\}, d),$ which in [@sherstov09hshs] were sharpened and complemented with matching lower bounds to the following effect:

\[thm:R+sign\] Let $n,d$ be positive integers, $R=R^+(\{\pm1,\pm2,\dots,\pm n\},d).$ For $1\leq d\leq\log n,$ $$\begin{aligned}
   \exp{\left\{-\Theta{\left(\frac1{n^{1/(2d)}}\right)}\right\}}
\leq R < \exp{\left\{-\frac1{n^{1/d}}\right\}}. \end{aligned}$$ For $\log n < d <n,$ $$\begin{aligned}
   R=\exp{\left\{-\Theta{\left(\frac{d}{\log (2n/d)}\right)}\right\}}.\end{aligned}$$ For $d\geq n,$ $$\begin{aligned}
   R=0.\end{aligned}$$

Theorem \[thm:R+sign\] 