Boundary asymptotics for rational orthonormal functions and Schur analysis

Given a positive measure on the unit circle with Dini-smooth nonvanishing density, and a hyperbolically non-separated sequence of points in the unit disk that may tend to the unit circle, we show a Szegő-type theorem for the initial coecient of orthonormal rational functions with poles at the reflection of these points.
This provides one with a convergence property of the multipoint Schur algorithm in the $L^2$-hyperbolic metric to Dini-smooth strictly Schur functions. It also yield some asymptotic behaviour for the orthonormal polynomials with respect to the previous measure with varying weight the squared modulus of the polynomial vanishing at the first $n$ points. Note the latter tend to the circle.