Orthogonal polynomials associated with equilibrium measures on $\mathbb{R}$

TL;DR

This paper investigates the behavior of orthogonal polynomials associated with equilibrium measures on real sets, establishing bounds on their norms and conditions for unboundedness, with detailed analysis for finite interval unions.

Contribution

It provides new bounds on the Hilbert norms of orthogonal polynomials related to equilibrium measures and characterizes when these norms become unbounded.

Findings

Lower bound of the polynomial norms by the capacity raised to the power n

Conditions under which the norms are unbounded

Detailed results for finite unions of intervals

Abstract

Let $K$ be a non-polar compact subset of $\mathbb{R}$ and $\mu_K$ denote the equilibrium measure of $K$. Furthermore, let $P_n\left(\cdot, \mu_K\right)$ be the $n$-th monic orthogonal polynomial for $\mu_K$. It is shown that $\|P_n\left(\cdot, \mu_K\right)\|_{L^2(\mu_K)}$, the Hilbert norm of $P_n\left(\cdot, \mu_K\right)$ in $L^2(\mu_K)$, is bounded below by $\mathrm{Cap}(K)^n$ for each $n\in\mathbb{N}$. A sufficient condition is given for $\displaystyle\left(\|P_n\left(\cdot;\mu_K\right)\|_{L^2(\mu_K)}/\mathrm{Cap}(K)^n\right)_{n=1}^\infty$ to be unbounded. More detailed results are presented for sets which are union of finitely many intervals.