# Orthogonal Rational Functions

@inproceedings{Bultheel2009OrthogonalRF, title={Orthogonal Rational Functions}, author={Adhemar Bultheel and Pablo Gonz{\'a}lez-Vera and Erik Hendriksen and Olav Nj{\aa}stad}, booktitle={Cambridge monographs on applied and computational mathematics}, year={2009} }

List of symbols Introduction 1. Preliminaries 2. The fundamental spaces 3. The kernel functions 4. Recurrence and second kind functions 5. Para-orthogonality and quadrature 6. Interpolation 7. Density of the rational functions 8. Favard theorems 9. Convergence 10. Moment problems 11. The boundary case 12. Some applications Conclusion Bibliography Index.

## 136 Citations

On the existence of para-orthogonal rational functions on the unit circle

- Mathematics
- 2010

Abstract Similar as in the classical case of polynomials, as is known, para-orthogonal rational functions on the unit circle can be used to obtain quadrature formulas of Szegő-type to approximate…

Explicit orthogonal polynomials for reciprocal polynomial weights on (

- Mathematics
- 2008

Let S be a polynomial of degree 2n + 2, that is, positive on the real axis, and let w = 1/S on (―∞, ∞). We present an explicit formula for the nth orthogonal polynomial and related quantities for the…

A Favard theorem for rational functions with complex poles

- Mathematics
- 2008

Let {φn} be a sequence of rational functions with arbitrary complex poles, generated by a certain three-term recurrence relation. In this paper we show that under some mild conditions the rational…

RATIONAL KRYLOV SEQUENCES AND ORTHOGONAL RATIONAL FUNCTIONS

- Computer Science
- 2008

The relationship between spectral decomposition, orthogonal rational functions and the rational Lanczos algorithm, based on a simple identity for rational Krylov sequences, is studied.

Holomorphic functions associated with indeterminate rational moment problems

- MathematicsJ. Comput. Appl. Math.
- 2015

CHRISTOFFEL FUNCTIONS AND UNIVERSALITY LIMITS FOR ORTHOGONAL RATIONAL FUNCTIONS

- Mathematics
- 2011

We establish limits for Christoffel functions associated with orthogonal rational functions, whose poles remain a fixed distance away from the interval of orthogonality [-1, 1], and admit a suitable…

Multipoint Schur algorithm and orthogonal rational functions, I: Convergence properties

- Mathematics
- 2008

Classical Schur analysis is intimately related to the theory of orthogonal polynomials on the circle. We investigate the connection between multipoint Schur analysis and orthogonal rational…

Orthogonal rational functions and rational modifications of a measure on the unit circle

- MathematicsJ. Approx. Theory
- 2009

## References

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A study of Orthogonal rational functions, which generalize the orthogonal Szegö polynomials, and focuses on the functions of the second kind which are natural generalizations of the corresponding polynmials.

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We shall consider nested spaces Ln, n = 0,1,2,... of rational functions with n prescribed poles outside the unit disk of the complex plane. We study orthogonal basis functions of these spaces for a…

On characterization theorems for measures associated with orthogonal systems of rational functions on the unit circle

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Moment Theory, Orthogonal Polynomials, Quadrature, and Continued Fractions Associated with the unit Circle

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Etude des relations entre la theorie des moments, les polynomes orthogonaux, la quadrature et les fractions continues associees au cercle-unite. Revue d'ensemble des resultats generaux

On a Special Laurent-Hermite Interpolation Problem

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We present a recursive algorithm for the construction of a rational approxi mation for a given Laurent series which in a certain sense interpolates at the zeros of its numerator. If certain symmetry…

Schur recursions, error formulas, and convergence of rational estimators for stationary stochastic sequences

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An exact and approximate realization theory for estimation and model filters of second-order stationary stochastic sequences is presented, and how the techniques presented constitute a generalization of many aspects of the Levinson-Szego theory of partial realizations is shown.

On the role of the Nevanlinna–Pick problem in circuit and system theory†

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The paper is concerned with applications of the standard Nevanlinna–Pick problem in various technical domains, namely the following: interpolation by reflectance functions, polynomial stability…