Asymptotics for orthogonal polynomials off the circle (Q2570873)

Let \[ \nu =\frac{\mu}{2\pi} +\sum_{k=1}^\infty A_k \delta(z-z_k) \] be a positive measure such that \(\mu\) is supported on the unit circle \(\mathbb T=\{ | z| =1\}\) and is absolutely continuous with respect to the Lebesgue measure on \(\mathbb T\), and the mass points \(z_k\) and the corresponding weights \(A_k>0\) satisfy \[ | z_k| >1\,, \quad \sum_{k=1}^\infty (| z_k| -1)<\infty\,, \quad \sum_{k=1}^\infty A_k <\infty\,. \tag{1} \] Then the sequence of orthonormal polynomials \(\Phi_n(z)=\gamma_n z^n+\dots\), \(\gamma _n>0\), can be defined, satisfying \[ \int \Phi_m(z) \overline{\Phi_n(z)}\, d\nu(z)=\delta_{mn}\,. \] The main goal of this paper is the behavior of the leading coefficients and of the orthogonal polynomials as \(n\to \infty\). Under an additional assumption on the mass points \(z_k\) and the weights \(A_k\) the limit of both \(\{ \gamma_n \}\) and \(\{ \Phi_n(z)/z^n\}\) when \(| z| >1\) is established. The additional assumption is not straightforward to verify, and it originates in the method of proof, which compares the \(\Phi_n\) with the orthogonal polynomials corresponding to measures obtained from \(\nu\) by truncating the discrete part at a certain (finite) number of mass points. It should be mentioned that recently the same results have been obtained by \textit{F. Nazarov, A. Volberg} and \textit{P. Yuditskii} [preprint in arXiv:math-ph/0510012] without any additional assumption on the discrete part of \(\nu\) beside (1).