Asymptotics of Chebyshev rational functions with respect to subsets of the real line

Abstract

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 $L^{\infty }$ norm. We study Chebyshev and residual extremal problems for rational functions with real poles with respect to subsets of $\overline{\mathbb{R}}$. 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.

33 pages