Consider $f\colon [a, b]\to\mathbb{R}$, with $a,b\in\mathbb{R}\cup \{\pm\infty\}$ and $a<b$, and let $f_n:=\arg\min_{\text{Polynomial p of degree n}} \sup_{x\in (a,b)} |f(x)-p(x)|$.
Is there any research on the behavior of $f_n$ outside of $[a,b]$? For example, if $f$ is (the restriction to $[a,b]$) of a meromorphic function, do $f_n$ converge to $f$ uniformly on all compact sets that don't include poles? (I convinced myself that this is true for holomorphic functions).