Asymptotics of orthogonal polynomials and point perturbation on the unit circle (Q979045)

Suppose \(d\mu\) is a probability measure on the unit circle, \(\partial\mathbb D=\{z\in\mathbb C:|z|=1\}\) and \[ \langle f,g \rangle=\int_{|z|=1}\overline{f(e^{i\theta})}g(e^{i\theta})\, d\mu(\theta) \] is an inner product. Let \((\Phi_n(z,d\mu))_{n\in\mathbb N}\) be the family of monic orthogonal polynomials associated with the measure \(d\mu\), obtained by orthogonalization of the polynomials \(1,z,z^2,\dotsc\). The normalized family of \(\Phi_n (z,d\mu)\) is denoted by \((\varphi_n(z, d\mu))_{n\in\mathbb N}\). Let \(\Phi^*_n(z)=z^n\overline{\Phi_n(z)}\) then \(\Phi_{n+1}(z)=z \Phi_n(z)-\bar\alpha_n \Phi^*_n(z)\), where \(\alpha_n\) is known as the \(n\)-th Verblunsky coefficient. Let \(d\nu=(1-\gamma)d\mu+\gamma \delta_\omega\), where \(\xi=e^{i\omega}\in\partial\mathbb D\) and \(\gamma\in(0,1)\). The goal of the paper is to investigate \(\alpha_n(d\nu)\). The class of measures with asymptotically periodic Verblunsky coefficients of \(p\)-type bounded variation (i.e., given a periodic sequence \(\beta_n\) of period \(p\), \(\lim _{n\to\infty}(\alpha_n-\beta_n)=0\) and \(\sum_{n=0}^\infty| \alpha_{n+1}-\alpha_n|<\infty\)) is considered. When the family \((\alpha_n)_{n\in\mathbb N}\) is asymptotically constant and of bounded variation in Theorem 2.1 an asymptotic formula for the orthogonal polynomials \(\varphi_n(z)\) in the gap of the spectrum is given. Also it is proved that the perturbation terms converges and the perturbation \(\Delta_n (\xi)\) is of bounded variation. In Theorem 2.2 the method used in Theorem 2.1 is generalized to the case of asymptotically periodic Verblunsky coefficients. In Theorem 2.3 the case when \((\alpha_n)_{n\in\mathbb N}\) is not necessarily of bounded variation is considered. In Corollary 2.1, with the help of Theorem 2.3, it is proved that if \(\alpha_n=L+c_n\), where \(L<0\), \(c_n\in\mathbb R\) and \(c_n\to0\), then \(\Delta_n(1)=-2L-2c_n+\text{o}(c_n)\).