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

Xu, Xu; Zhu, Laiyi

Rocky Mt. J. Math. 48, No. 3, 1019-1030 (2018).

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.

Reviewer: Olga M. Katkova (Boston)

MSC:

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

Keywords:

orthogonal rational functions; Nevanlinna representation; Carathéodory functions; Riesz-Herglotz kernel; Poisson kernel; interpolation by rational functions

References:

[1]	A. Bultheel, P. González-Vera, E. Hendriksen and O. Njåstad, Orthogonal rational functions, Cambr. Mono. Appl. Comp. Math. 5, Cambridge, 1999. · Zbl 0923.42017
[2]	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. · Zbl 1049.42012
[3]	—-, Orthogonal rational functions on the half line with poles in \([-∞,0]\), J. Comp. Appl. Math. 179 (2005), 121-155. · Zbl 0949.42022
[4]	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. · Zbl 1229.42027
[5]	—-, Orthogonal rational functions, associated rational functions and functions of the second kind, Proc. World Congr. Eng. 2 (2008), 838-843.
[6]	—-, Recurrence and asymptotics for orthogonal rational functions on an interval, IMA. J. Numer. Anal 29 (2009), 1-23. · Zbl 1159.65025
[7]	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. · Zbl 1214.30021
[8]	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. · Zbl 1117.30030
[9]	G. Freud, Orthogonal polynomials, Pergamon Press, Oxford, 1971. · Zbl 0226.33014
[10]	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. · Zbl 0763.42016
[11]	—-, 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. · Zbl 0790.42011
[12]	—-, On the convergence of the rational interpolation approximant of the Carathéodory functions, J. Comp. Appl. 54 (1994), 371-376. · Zbl 0818.41011
[13]	—-, On the orthogonal rational functions with arbitrary poles and interpolation properties, J. Comp. Appl. 60 (1993), 347-355.