Orthogonal rational functions on the extended real line and analytic on the upper half plane
type: Item
instance of: scholarly article
title: Orthogonal rational functions on the extended real line and analytic on the upper half plane
zbMATH Open document ID: 1402.30007
DOI: 10.1216/RMJ-2018-48-3-1019
author: Xu Xu
author: Laiyi Zhu
published in: Rocky Mountain Journal of Mathematics
publication date: 2018-08-14T00:00:00Z
full work available at URL: https://projecteuclid.org/euclid.rmjm/1533230838
review text: 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.
reviewed by: Olga M. Katkova
Mathematics Subject Classification ID: 30C15
Mathematics Subject Classification ID: 41A20
Mathematics Subject Classification ID: 42C05
Mathematics Subject Classification ID: 30C20
Mathematics Subject Classification ID: 30C40
zbMATH DE Number: 6917361
zbMATH Keywords: orthogonal rational functions
zbMATH Keywords: Nevanlinna representation
zbMATH Keywords: Carathéodory functions
zbMATH Keywords: Riesz-Herglotz kernel
zbMATH Keywords: Poisson kernel
zbMATH Keywords: interpolation by rational functions
MaRDI profile type: Publication
cites work: Q4223056
cites work: Orthogonal rational functions and quadrature on the real half line.
cites work: Regularity of orthogonal rational functions with poles on the unit circle
cites work: Q3107507
cites work: Recurrence and asymptotics for orthonormal rational functions on an interval
cites work: An extension of the associated rational functions on the unit circle
cites work: An extended relation between orthogonal rational functions on the unit circle and the interval \([ - 1,1]\)
cites work: A class of orthogonal polynomials
cites work: On characterization theorems for measures associated with orthogonal systems of rational functions on the unit circle
cites work: On orthogonal systems of rational functions on the unit circle and polynomials orthogonal with respect to varying measures
cites work: On the convergence of the rational interpolation approximant of Carathéodory functions
cites work: On the orthogonal rational functions with arbitrary poles and interpolation properties
description: scientific article; zbMATH DE number 6917361