Orthogonal polynomials associated with equilibrium measures on ℝ

2016/03/24 by Alpan, Gökalp
#31A15 #42C05 #Classical Analysis and ODEs (math.CA) #FOS: Mathematics

paper · doi:10.48550/arxiv.1603.07705

Abstract

Let K be a non-polar compact subset of 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(⋅, μ_K)‖_L²(μK), the Hilbert norm of P_n(⋅, μ_K) in L²(μ_K), is bounded below by Cap(K)ⁿ for each n∈ℕ. A sufficient condition is given for (‖P_n(⋅;μ_K)‖_L²(μK)/Cap(K)ⁿ)_n=1^∞ to be unbounded. More detailed results are presented for sets which are union of finitely many intervals.

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