Asymptotics of Chebyshev Polynomials, V. Residual Polynomials

Abstract

We study residual polynomials, $R_{x_0,n}^{(\mathfrak{e})}$, $\mathfrak{e}\subset \mathbb{R}$, $x_0\in \mathbb{R}\setminus \mathfrak{e}$, which are the degree at most $n$ polynomials with $R(x_0)=1$ that minimize the $\sup$ norm on $\mathfrak{e}$. New are upper bounds on their norms (that are optimal in some cases) and Szegő--Widom asymptotics under fairly general circumstances. We also discuss several illuminating examples and some results in the complex case.

30 pages