Asymptotics of Chebyshev Polynomials, V. Residual Polynomials

TL;DR

This paper investigates the asymptotic behavior of residual Chebyshev polynomials, providing new bounds and Szeg\

Contribution

It introduces new upper bounds and asymptotic results for residual polynomials, extending understanding of their behavior in real and complex cases.

Findings

Derived optimal upper bounds on residual polynomial norms

Established Szeg\

discussed examples illustrating theoretical results

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 Szeg\H{o}--Widom asymptotics under fairly general circumstances. We also discuss several illuminating examples and some results in the complex case.