Rocky Mountain Journal of Mathematics

Orthogonal rational functions on the extended real line and analytic on the upper half plane

Xu Xu and Laiyi Zhu

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Abstract

Let $\{\alpha _k\}_{k=1}^\infty$ be an arbitrary sequence of complex numbers in the upper half plane. We generalize the orthogonal rational functions $\phi _n$ based upon those points and obtain the Nevanlinna measure, together with the Riesz and Poisson kernels, for Caratheodory functions $F(z)$ on the upper half plane. Then, we study the relation between ORFs and their functions of the second kind as well as their interpolation properties. Further, by using a linear transformation, we generate a new class of rational functions and state the necessary conditions for guaranteeing their orthogonality.

Article information

Source
Rocky Mountain J. Math., Volume 48, Number 3 (2018), 1019-1030.

Dates
First available in Project Euclid: 2 August 2018

Permanent link to this document
https://projecteuclid.org/euclid.rmjm/1533230838

Digital Object Identifier
doi:10.1216/RMJ-2018-48-3-1019

Mathematical Reviews number (MathSciNet)
MR3835585

Zentralblatt MATH identifier
06917361

Subjects
Primary: 30C15: Zeros of polynomials, rational functions, and other analytic functions (e.g. zeros of functions with bounded Dirichlet integral) {For algebraic theory, see 12D10; for real methods, see 26C10} 30C20: Conformal mappings of special domains 41A20: Approximation by rational functions 42C05: Orthogonal functions and polynomials, general theory [See also 33C45, 33C50, 33D45]

Keywords
Orthogonal rational functions upper half plane Nevanlinna representation interpolation

Citation

Xu, Xu; Zhu, Laiyi. Orthogonal rational functions on the extended real line and analytic on the upper half plane. Rocky Mountain J. Math. 48 (2018), no. 3, 1019--1030. doi:10.1216/RMJ-2018-48-3-1019. https://projecteuclid.org/euclid.rmjm/1533230838


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