## Rocky Mountain Journal of Mathematics

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

#### 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

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

#### 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

#### References

• A. Bultheel, P. González-Vera, E. Hendriksen and O. Njåstad, Orthogonal rational functions, Cambr. Mono. Appl. Comp. Math. 5, Cambridge, 1999.
• A. Bultheel, P. González-Vera, E. Hendriksen and O. Njåstad, Orthogonal rational functions and quadrature on the real half line, J. Complexity 19 (2003), 212–230.
• ––––, Orthogonal rational functions on the half line with poles in $[-\infty,0]$, J. Comp. Appl. Math. 179 (2005), 121–155.
• K. Deckers and A. Bultheel, Associated rational functions based on a three-term recurrence relation for orthogonal rational functions, IAENG Int. J. Appl. Math. 38 (2008), 214–222.
• ––––, Orthogonal rational functions, associated rational functions and functions of the second kind, Proc. World Congr. Eng. 2 (2008), 838–843.
• ––––, Recurrence and asymptotics for orthogonal rational functions on an interval, IMA. J. Numer. Anal 29 (2009), 1–23.
• K. Deckers, M. Cantero, L. Moral and L. Velázquesz, An extension of the associated rational functions on the unit circle, J. Approx. Th. 163 (2011), 524–546.
• K. Deckers, J. Van Deun and A. Bultheel, An extended relation between orthogonal rational functions on the unit circle and the interval $[-1,1]$, J. Math. Anal. Appl 334 (2007), 1260–1275.
• G. Freud, Orthogonal polynomials, Pergamon Press, Oxford, 1971.
• K. Pan, On characterization theorems for measures associated with orthogonal systems of rational functions on the unit circle, J. Approx. Th. 70 (1992), 265–272.
• ––––, On orthogonal systems of rational functions on the unit circle and polynomials orthogonal with respect to varying measures, J. Comp. Appl. 47 (1993), 313–332.
• ––––, On the convergence of the rational interpolation approximant of the Carathéodory functions, J. Comp. Appl. 54 (1994), 371–376.
• ––––, On the orthogonal rational functions with arbitrary poles and interpolation properties, J. Comp. Appl. 60 (1993), 347–355.