Orthogonal rational functions on the extended real line and analytic on the upper half plane. (English) Zbl 1402.30007

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.


30C15 Zeros of polynomials, rational functions, and other analytic functions of one complex variable (e.g., zeros of functions with bounded Dirichlet integral)
41A20 Approximation by rational functions
42C05 Orthogonal functions and polynomials, general theory of nontrigonometric harmonic analysis
30C20 Conformal mappings of special domains
30C40 Kernel functions in one complex variable and applications
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