Orthogonal Polynomials Associated with Equilibrium Measures on ℝ

Abstract

Let K be a non-polar compact subset of ℝ and μK denote the equilibrium measure of K. Furthermore, let Pn(⋅;μK) be the n-th monic orthogonal polynomial for μK. It is shown that ∥Pn(⋅;μK)∥L2(μK), the Hilbert norm of Pn(⋅;μK) in L2(μK), is bounded below by Cap(K)n for each n∈ ℕ. A sufficient condition is given for(∥Pn(⋅;μK)∥L2(μK)/Cap(K)n)n=1∞ to be unbounded. More detailed results are presented for sets which are union of finitely many intervals. © 2016, Springer Science+Business Media Dordrecht.