Boundary asymptotics for orthogonal rational functions on the unit circle (Q1591315)

For a sequence of points \(\alpha_k\) in a compact subset of the open unit disk, the authors consider the orthogonal rational functions (with prescribed poles) obtained by orthogonalizing the sequence \(1,z/\pi_1(z), z^2/\pi_2(z),\ldots,z^n/\pi_n(z)\), where \(\pi_n(z) = \prod_{j=1}^n(1-\bar{\alpha}_j z)\), with respect to a positive measure \(\mu\) on the unit circle. The main result (Theorem 1.1) is the asymptotic behaviour of these orthogonal rational functions on the closed unit disk, with a rate of convergence. The condition on the measure \(\mu\) is that it is absolutely continuous with a weight \(w\) which is bounded away from 0 and \(\infty\), and this weight satisfies a Lipschitz-Dini condition. Furthermore, the counting measures \(\nu_n\) for the points \(\{\alpha_1,\ldots,\alpha_n\}\) are supposed to converge weakly to a measure on a compact subset of the open unit disk. The proof is along the lines of Bernstein-Szegő, and consists of approximating \(1/w\) by trigonometric rational functions with prescribed poles (Theorem 2.1) and by obtaining a rational equivalent of the Bernstein-Szegő polynomials (Theorem 3.1).