Some results about numerical quadrature on the unit circle

González-Vera, Pablo; Santos-León, J. C.; Njåstad, Olav

doi:10.1007/bf02124749

Cited by 43 publications

(42 citation statements)

References 9 publications

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“…Condition (4) is satisfied by Szegö quadrature formulas (3). Indeed, let the moments m k be given by…”

Section: Definitionmentioning

confidence: 99%

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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“…Condition (4) is satisfied by Szegö quadrature formulas (3). Indeed, let the moments m k be given by…”

Section: Definitionmentioning

confidence: 99%

Error bounds for interpolatory quadrature rules on the unit circle

Santos-León

2000

Math. Comp.

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“…2.1]). Such results in a setting based on linear algebra and also without the use of the name para orthogonal appear, even earlier than in [12], in Gragg [13]. However, the name para orthogonal polynomials for S n ðzÞ s n S Ã n ðzÞ, where js n j 1 and S n are OPUC, is due to Jones et al [17].…”

Section: Para-orthogonal Polynomials From Kernel Polynomialsmentioning

confidence: 98%

“…Perhaps the first reference that explicitly brings the connection between CD kernels and para orthogonal polynomials is González Vera et al [12] (see also [3, Thm. 2.1]).…”

Section: Para-orthogonal Polynomials From Kernel Polynomialsmentioning

confidence: 99%

Sieved para-orthogonal polynomials on the unit circle

Marcellán

Ranga

2014

Applied Mathematics and Computation

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“…The polynomials Φ n have all their zeros in D so they cannot be used to construct on T an analog of the Gauss-Jacobi rule. Instead, polynomials called para-orthogonal (because of deficiencies in their orthogonality properties) are considered (see [11,14]). Though the introduction of para-orthogonal polynomials is generally traced back to [14], we have noticed that they appear previously in Theorem III of [10] by Ya.…”

Section: Stieltjes-type Polynomialsmentioning

confidence: 99%

Stieltjes-type polynomials on the unit circle

Ysern¹,

Lagomasino

Reichel

2008

Math. Comp.

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Abstract. Stieltjes-type polynomials corresponding to measures supported on the unit circle T are introduced and their asymptotic properties away from T are studied for general classes of measures. As an application, we prove the convergence of an associated sequence of interpolating rational functions to the corresponding Carathéodory function. In turn, this is used to give an estimate of the rate of convergence of certain quadrature formulae that resemble the Gauss-Kronrod rule, provided that the integrand is analytic in a neighborhood of T.

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Some results about numerical quadrature on the unit circle

Cited by 43 publications

References 9 publications

Error bounds for interpolatory quadrature rules on the unit circle

Error bounds for interpolatory quadrature rules on the unit circle

Sieved para-orthogonal polynomials on the unit circle

Stieltjes-type polynomials on the unit circle

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