Asymptotic Behavior of Orthogonal Polynomials Corresponding to Measure with Discrete Part Off the Unit Circle

AbstractFor a positive measure μ on the unit circle (Γ) in the complex plane, m points zj off Γ and m positive numbers Aj, j = 1, 2,..., m, we investigate the asymptotic behavior of orthonormal polynomials Φn(z) corresponding to d μ/2π + Σmj=1Aj δzj, where δz denotes the unit measure supported at point z. Our main result is the relative asymptotics of Φn(z) with respect to the orthonormal polynomial corresponding to d μ/(2 π) off and on Γ