Orthogonal rational functions on the extended real line and analytic on the upper half plane
Let \(\{\alpha_k\}_{k=1}^{\infty}\) be an arbitrary sequence of complex numbers in the upper half plane \( U=\{z: \text{Im}\;z>0\}\). The authors consider the rational function spaces \(\mathcal{L}_n=\mathcal{L}\{\alpha_1, \ldots, \alpha_n\}\), with poles \(\overline{\alpha_k}, \;k=1,2, \ldots, n, \;n\geq1.\) In the paper the Nevanlinna measure \(\tilde{\mu}\), together with the Riesz kernel \(D(t, z)\) and Poisson kernel \(P(t, z)\), for Carathéodory functions \(F\) (holomorphic in \( U \) such that \(\text{Re}\;F(z)>0,\;z \in U\)) are obtained. The authors associate with the normalized Nevanlinna representation a Hermitian positive definite linear inner product \(\mathcal{H}_F \{f\}=\int_ \mathbb{R} f(t) \;d \tilde{\mu }( t) ,\) \( (f, g)=\mathcal{H} \{fg_{\ast}\},\) where \(g_{\ast}(z)=\overline{g(\bar{z})},\) and construct the orthogonal rational functions \(\phi_n \in \mathcal{L}_n \setminus \{0\}.\) Then they study the relation between orthogonal rational functions \(\phi_n\) and their functions of the second kind \(\psi_n(z)=\mathcal{H}_F\{ D(t,z)[\phi_n(t)-\phi_n(z)]\}+\mathcal{H}_F\{\phi_n(t)\}, \;n \geq 0.\) The authors show that \(-\psi_n(z) / \phi_n(z)\) interpolate a Carathéodory function \(F(z)\) at the points \(\{\alpha_k\}_{k=1}^{n-1}\) while \(\psi_{n \ast }(z) / \phi_{n \ast}(z)\) interpolate \(F(z)\) at the points \(\{\alpha_k\}_{k=1}^{n}.\) Further, by using a linear transformation, they generate a new class of rational functions and state the necessary conditions for guaranteeing their orthogonality.
scientific article; zbMATH DE number 6917361