Orthogonal Polynomials Associated with Equilibrium Measures on ℝ $\mathbb {R}$

Abstract

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