### Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 13 (2017), 090, 49 pages      arXiv:1707.09748      https://doi.org/10.3842/SIGMA.2017.090
Contribution to the Special Issue on Orthogonal Polynomials, Special Functions and Applications (OPSFA14)

### Orthogonal Rational Functions on the Unit Circle with Prescribed Poles not on the Unit Circle

Adhemar Bultheel a, Ruyman Cruz-Barroso b and Andreas Lasarow c
a) Department of Computer Science, KU Leuven, Belgium
b) Department of Mathematical Analysis, La Laguna University, Tenerife, Spain
c) Fak. Informatik, Mathematik & Naturwissenschaften, HTWK Leipzig, Germany

Received August 01, 2017, in final form November 20, 2017; Published online December 03, 2017

Abstract
Orthogonal rational functions (ORF) on the unit circle generalize orthogonal polynomials (poles at infinity) and Laurent polynomials (poles at zero and infinity). In this paper we investigate the properties of and the relation between these ORF when the poles are all outside or all inside the unit disk, or when they can be anywhere in the extended complex plane outside the unit circle. Some properties of matrices that are the product of elementary unitary transformations will be proved and some connections with related algorithms for direct and inverse eigenvalue problems will be explained.

Key words: orthogonal rational functions; rational Szegő quadrature; spectral method; rational Krylov method; AMPD matrix.

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