Asymptotics of Chebyshev Rational Functions with Respect to Subsets of the Real Line

There is a vast theory of Chebyshev and residual polynomials and their asymptotic behavior. The former ones maximize the leading coefficient and the latter ones maximize the point evaluation with respect to an \(L^\infty \) $L^{\infty }$ norm. We study Chebyshev and residual extremal problems for rational functions with real poles with respect to subsets of \(\overline{{{\mathbb {R}}}}\) $\overline{\mathbb{R}}$ . We prove root asymptotics under fairly general assumptions on the sequence of poles. Moreover, we prove Szegő–Widom asymptotics for sets which are regular for the Dirichlet problem and obey the Parreau–Widom and DCT conditions.