Asymptotic behavior of orthogonal polynomials corresponding to measure with discrete part off the unit circle (Q1335042)

The paper is devoted to the study of the asymptotic behavior of orthonormal polynomials on the unit circle. More exactly: Given a positive measure \(\mu\) on the unit circle \(\Gamma\), \(m\) points \(z_ j\) off \(\Gamma\), and \(m\) positive numbers \(A_ j\), \(j=1,\dots,m\), the asymptotic behavior of the orthonormal polynomials \(\Phi_ n(z)\) with respect to the measure \(\mu+ \sum^ m_{j=1} A_ j\delta_{z_ j}\) is investigated (\(\delta_ z\) the Dirac measure at \(z\)). The main result is concerned with relative asymptotics of \(\Phi_ n(z)\) with respect to the polynomials orthonormal to the measure \(\mu\). Relative asymptotics means that the ratio of both systems of polynomials is considered.