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 $\Vert P_n\left(\cdot ,{\mu }_K\right){\Vert }_{L^2({\mu }_K)}$, 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 ${\left(\Vert P_n\left(\cdot ;{\mu }_K\right){\Vert }_{L^2({\mu }_K)}/\mathrm{Cap}(K{)}^n\right)}_{n=1}^{\infty }$ to be unbounded. More detailed results are presented for sets which are union of finitely many intervals.