Orthogonal rational functions and modified approximants
This paper studies certain rational approximants to the function \[ F_\mu(z)= \int^\pi_{- \pi} {t+ z\over t- z} d\mu(\theta),\quad(t= e^{i\theta}), \] where \(\mu\) is a positive Borel measure on the unit circle in the complex plane. These approximants are constructed in terms of orthogonal rational functions with fixed poles. In fact, given a sequence \(\alpha_0= 0, \alpha_1, \alpha_2,\dots\) of complex numbers in \(\{|z |< 1\}\), the Gramm-Schmidt process with respect to \(\mu\) applied to the finite Blaschke products \[ B_0= 1,\quad B_n(z)= \prod^n_{k= 0} {\overline{\alpha_k}\over |\alpha_k|} {\alpha_k- z\over 1- \overline{\alpha_k} z},\quad n= 1,2,\dots, \] leads to the orthonormal sequence \(\{\phi_n\}\) and to the sequence of associated (second kind) rational functions \(\{\psi_n\}\). A conjugation \(f^*\) in the class \(\text{span}\{B_m, m= 1,\dots, n, n\in \mathbb{N}\}\) is defined. The sequences \(\{\phi_n\}\), \(\{\phi^*_n\}\), and \(\{\psi^*_n\}\) allow to construct the modified Padé-type approximants to \(F_\mu\). Besides, some results concerning related quadrature formulas are established.
scientific article; zbMATH DE number 871274