On the norms and roots of orthogonal polynomials in the plane and $L^p$-optimal polynomials with respect to varying weights

TL;DR

This paper studies the roots and asymptotic behavior of $L^p$-optimal weighted polynomials in the complex plane, revealing their roots cluster near the equilibrium measure's support and providing norm asymptotics.

Contribution

It extends the analysis of weighted orthogonal polynomials to all $p$-norms, including the case $p=inite$, and describes root distribution and norm asymptotics.

Findings

Most roots lie near the equilibrium measure's support.

Asymptotics for the $n$th roots of the $L^p$ norms are derived.

The case $p=2$ recovers classical orthogonal polynomial results.

Abstract

For a measure on a subset of the complex plane we consider $L^p$-optimal weighted polynomials, namely, monic polynomials of degree $n$ with a varying weight of the form $w^n = {\rm e}^{-n V}$ which minimize the $L^p$-norms, $1 \leq p \leq \infty$. It is shown that eventually all but a uniformly bounded number of the roots of the $L^p$-optimal polynomials lie within a small neighborhood of the support of a certain equilibrium measure; asymptotics for the $n$th roots of the $L^p$ norms are also provided. The case $p=\infty$ is well known and corresponds to weighted Chebyshev polynomials; the case $p=2$ corresponding to orthogonal polynomials as well as any other $1\leq p <\infty$ is our contribution.