Strong and weak convergence of orthogonal systems of rational functions on the unit circle
Let \(\{\alpha_ i\}_{i=1}^ \infty\) with all \(\alpha_ i\in\Delta:=\{z\in\mathbb{C}: | z|<1\}\), and let \(d\mu\) be a finite positive Borel measure on the unit circle \(\partial\Delta\) whose support contains infinitely many points. Then the sequence \[ \varphi_ 0(z):=1,\qquad\varphi_ n(z):=\prod_{k=1}^ \infty{|\alpha_ k|\over \alpha_ k}{\alpha_ k-z\over 1-\overline\alpha_ kz},\quad n=1,2,\dots, \] of rational functions, where \(|\alpha_ k|/\alpha_ k:=-1\) if \(\alpha_ k=0\), is well defined, and orthogonalization of \(\{\varphi_ n\}_{n=0}^ \infty\) with respect to \(d\mu(\theta)\) on \(\partial\Delta\) gives a system \(\{\phi_ n(z)\}_{n=0}^ \infty\) of orthonormal rational functions on \(\partial\Delta\). For this system the author generalizes results on strong and weak convergence of \textit{A. Máté}, \textit{P. Navai} and \textit{V. Totik} [Am. J. Math. 109, 239-282 (1987; Zbl 0633.42008)]. He also establishes characterization theorems for Erdős measures by generalizing results of \textit{X. Li} and \textit{E. B. Saff} [J. Approximation Theory 63, No. 2, 191- 197 (1990; Zbl 0758.42015)].
scientific article; zbMATH DE number 218434
scientific article