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

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.