Orthogonal rational functions and quadrature on the unit circle

Bultheel, Adhemar; González-Vera, Pablo; Hendriksen, Erik; Njåstad, Olav

doi:10.1007/bf02141920

Cited by 28 publications

(29 citation statements)

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“…An n-point Szegö quadrature formula is of interpolatory type ( [2]) in Λ −pn,qn for any p n and q n nonnegative integers satisfying p n + q n = n − 1. The following theorem is proved in [1] for the general case where the basis functions for the approximation is the set of rational functions with prescribed poles, which includes the Laurent polynomials as a particular case.…”

Section: Definitionmentioning

confidence: 99%

“…Error bounds for Szegö quadrature formulas of analytic functions were given in [2], and for the particular case of integrals that represent Carathéodory functions and real parts of such integrals in [7]. In [8] error bounds for interpolatory integration rules were studied for analytic functions.…”

Section: Definitionmentioning

confidence: 99%

“…We denote by Λ p,q the linear space of all functions of the form with respect to the inner product (2). The sequence {ρ n } ∞ 0 is the sequence of Szegö polynomials with respect to the distribution function ψ.…”

Section: Introductionmentioning

confidence: 99%

“…In [2] Bultheel, González-Vera, Hendriksen and Njåstad proved that the Szegö quadrature process converges as n tends to infinity to I(f ), for all integrable functions f on T with respect to the measure dψ. They also introduced the so-called interpolatory rules on the unit circle.…”

Section: Introductionmentioning

confidence: 99%

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Error bounds for interpolatory quadrature rules on the unit circle

Santos-León

2000

Math. Comp.

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Abstract. For the construction of an interpolatory integration rule on the unit circle T with n nodes by means of the Laurent polynomials as basis functions for the approximation, we have at our disposal two nonnegative integers pn and qn, pn + qn = n − 1, which determine the subspace of basis functions. The quadrature rule will integrate correctly any function from this subspace. In this paper upper bounds for the remainder term of interpolatory integration rules on T are obtained. These bounds apply to analytic functions up to a finite number of isolated poles outside T. In addition, if the integrand function has no poles in the closed unit disc or is a rational function with poles outside T , we propose a simple rule to determine the value of pn and hence qn in order to minimize the quadrature error term. Several numerical examples are given to illustrate the theoretical results.

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Section: Definitionmentioning

confidence: 99%

Section: Definitionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Error bounds for interpolatory quadrature rules on the unit circle

Santos-León

2000

Math. Comp.

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“…The following result about interpolatory quadrature formulas in the rational case, can be found in [6]. …”

Section: The Computation Of Rational Quadrature Formulasmentioning

confidence: 99%

A matricial computation of rational quadrature formulas on the unit circle

Bultheel

Cantero

2008

Numer Algor

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A matricial computation of quadrature formulas for orthogonal rational functions on the unit circle, is presented in this paper. The nodes of these quadrature formulas are the zeros of the para-orthogonal rational functions with poles in the exterior of the unit circle and the weights are given by the corresponding Christoffel numbers. We show how these nodes can be obtained as the eigenvalues of the operator Möbius transformations of Hessenberg matrices and also as the eigenvalues of the operator Möbius transformations of five-diagonal matrices, recently obtained. We illustrate the preceding results with some numerical examples. Abstract A matricial computation of quadrature formulas for orthogonal rational functions on the unit circle, is presented in this paper. The nodes of these quadrature formulas are the zeros of the para-orthogonal rational functions with poles in the exterior of the unit circle and the weights are given by the corresponding Christoffel numbers. We show how these nodes can be obtained as the eigenvalues of the operator Möbius transformations of Hessenberg matrices and also as the eigenvalues of the operator Möbius transformations of five-diagonal matrices, recently obtained. We illustrate the preceding results with some numerical examples.

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