Boundary asymptotics for orthogonal rational functions on the unit circle

Let w(t) be a positive weight function on the unit circle of the complex plane. For a sequence of points {a_k:k=1...∞} included in a compact subset of the unit disk, we consider the orthogonal rational functions φn that are obtained by orthogonalization of the sequence {1,z/π_1,z^2/π_2,...} where π_k(z) = ∏_{j=1...k} (1-ã_jz), with respect to the inner product = (1/2π)∫_{t=-π...π} f(e^{it})g(e^{it})*w(t)dt. We discuss in this paper the behaviour of φ_n(t) for |t|=1 and n → ∞ under certain conditions. The main condition on the weight is that it satisfies a Lipschitz-Dini condition and that it is bounded away from zero. This generalizes a theorem given by Szegö in the polynomial case, that is when all a_k=0.status: publishe