Ratio and relative asymptotics of polynomials orthogonal on an arc of the unit circle
In this interesting and important paper, the authors obtain ratio asymptotics for polynomials orthogonal with respect to a measure supported on an arc \(\gamma\) of the unit circle. Thus let \(\sigma\) be a finite positive Borel measure supported on \(\gamma\), and let \[ \varphi_n(\zeta)= \alpha_n\zeta^n+\cdots,\;\alpha_n>0, \] denote the \(n\)th orthonormal polynomial for \(\sigma\), so that \[ \int_{| z|= 1}\varphi_n(\zeta)\overline{\varphi_m(\zeta)} d\sigma(\zeta)= \delta_{mn},\;m,n\geq 0. \] The authors prove that \[ \lim_{n\to\infty} {\alpha_{n+ 1}\over\alpha_n}= {1\over\text{cap}(\gamma)}, \] where \(\text{cap}(\gamma)\) denotes the logarithmic capacity of \(\gamma\), and that \[ \lim_{n\to\infty} {\varphi_{n+ 1}(\zeta)\over \varphi_n(\zeta)}= G(\zeta) \] uniformly on compact subsets of \(\mathbb{C}\setminus\gamma\), where \(G\) denotes the conformal map of \(\mathbb{C}\setminus\gamma\) onto the complement of the unit ball. This extends a classic result of Rahmanov for the case where \(\gamma\) was the unit circle. Several applications and related results are presented. The topic treated by the authors is currently being actively investigated. This paper is undoubtedly one of the foundation papers for all subsequent research.
scientific article; zbMATH DE number 1146509
scientific article