On the convergence of the rational interpolation approximant of Carathéodory functions (Q1347163)

Let \(d\mu\) denote a finite positive Borel measure on the unit circle with associated Carathéodory function \(F(z)= \int [u/(u- z)] d\mu(u)\). Given a triangular table of interpolation nodes \(\{z_{n, k}: | z_{n, k}|< 1\}_{k= 1}^ n\), \(n= 0,1,\dots\), the rational interpolants \(p_{n- 1}/q_ n\) of type \((n- 1,n)\) for \(F\) in the points \(\{z_{n, k}, 1/\overline z_{n, k}\}_{k= 1}^ n\) are considered. First it is shown that \(q_ n\) can be chosen to be the \(n\)th orthogonal polynomial with respect to the varying measure \(d\mu_ n(u)= d\mu(u)/ | w_ n(u)|^ 2\), where \(w_ n(u)= \prod_{k= 1}^ n (1- \overline z_{n, k}u)\). From this, expressions for \(p_{n- 1}\) and the error \(e_ n= q_ n F- p_{n- 1}\) can be obtained. Then it is shown that the interpolants converge to \(F\) locally uniformly outside the unit disk when \[ \lim_{n\to\infty} \sum_{i= 1}^ n (1-| z_{n, i}|^ 2)= \infty. \] Finally, it is shown that some knowledge about the distribution of the nodes \(z_{n, k}\) leads to corresponding knowledge of the rate of convergence.