Asymptotic behavior of orthogonal polynomials corresponding to a measure with infinite discrete part off a curve
In this paper, the author obtains Szegö type asymptotics for orthogonal polynomials associated with a positive Borel measure \(\alpha\) on a rectifiable Jordan curve \(E\) in the complex plane. The measure has the form \(\alpha= \beta+\gamma\), where \(\beta\) satisfies the relevant form of Szegö's condition on the curve, and \(\gamma\) is a pure jump measure. The mass points of \(\gamma\) lie in a countable sequence of points \(\{z_k\}^\infty_{k=1}\) outside \(E\); technical conditions are placed involving these points, the conformal map \(\Phi\) of the exterior of \(E\) onto the exterior of the unit ball, and the monic orthonormal polynomial for \(d\alpha\), \(P_n(d\alpha, z)=z^n+\cdots\). The author proves the asymptotic formula \[ P_n(d\alpha,z)= (\text{cap}(E)\Phi(z))^n[\varphi^*(z) B(z)+o(1)] \] uniformly on compact subsets of the exterior of \(E\). Here \(\text{cap}(E)\) is the logarithmic capacity of \(E\) and \(B(z)\) is a Blaschke product involving the mass points \(\{z_k\}^\infty_{k=1}\) and the conformal map \(\Phi\). Moreover, \(\varphi^*(z)\) is the solution of an extremal problem for the Hardy 2 space associated with the measure \(\beta\) and the exterior of \(E\).
Szegö type asymptotics for orthogonal polynomials
scientific article; zbMATH DE number 1019194