Boundary Asymptotics for Orthogonal Rational Functions on the Unit Circle

Abstract

Let w(θ) be a positive weight function on the unit circle of the complex plane. For a sequence of points { α _k } _{k = 1} ^∞ included in a compact subset of the unit disk, we consider the orthogonal rational functions φ _n that are obtained by orthogonalization of the sequence { 1, z / π₁, z ² / π₂, ... } where ${\pi }_k \left( {\rm Z} \right) = \prod\nolimits_{j^{ = 1} }^k {\left( {1 - \overline {\alpha } j{\rm Z}} \right)}$, with respect to the inner product $\left\langle {f,g} \right\rangle = \frac{1}{{2{\pi }}}\int_{{ - \pi }}^{\pi } {f\left( {{e}^{i\theta } } \right)} \overline {g\left( {{e}^{i\theta } } \right)} w\left( {\theta } \right){d\theta }$ In this paper we discuss 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 Szegő in the polynomial case, that is when all α _k = 0.