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

TL;DR

This paper investigates the asymptotic behavior of Chebyshev and residual rational functions with real poles on subsets of the real line, providing new root and Szegő–Widom asymptotics under general conditions.

Contribution

It establishes root asymptotics for rational extremal problems and proves Szegő–Widom asymptotics for regular sets satisfying specific conditions, extending classical theory.

Findings

Root asymptotics for rational extremal functions

Szegő–Widom asymptotics for regular sets

General conditions on pole sequences

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 Szeg\H{o}--Widom asymptotics for sets which are regular for the Dirichlet problem and obey the Parreau--Widom and DCT conditions.