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Zbl 1203.65054
Bultheel, A.; González-Vera, P.; Hendriksen, E.; Njåstad, O.
Rational quadrature formulas on the unit circle with prescribed nodes and maximal domain of validity. (English)
[J] IMA J. Numer. Anal. 30, No. 4, 940-963 (2010). ISSN 0272-4979; ISSN 1464-3642/e

The authors study quadrature formulas of the form $$\int_{-\pi}^{\pi} f(e^{i\theta})\, d\mu(\theta) \approx \sum_{k=1}^n \lambda_k\, f(z_k),$$ where $\mu$ is a positive Borel measure, $\lambda_k>0$ and $z_k$ are nodes on the complex unit circle. It is shown that the quadrature formulas can be chosen to be exact in certain subspaces of rational functions of dimension $2n$. Rational Szegö-Radau and Szegö-Lobatto quadrature formulas, where one or two nodes are prescribed, are characterized too. Thus this paper generalizes results of {\it C.~Jagels} and {\it L.~Reichel} [J. Comput. Appl. Math. 200, No.~1, 116--126 (2007; Zbl 1109.65027)] from trigonometric polynomials to rational functions.
[Manfred Tasche (Rostock)]

MSC 2000:: *65D32 Quadrature formulas (numerical methods)
41A55 Approximate quadratures

Keywords: rational quadrature formula; integration of rational functions; Szegö quadrature; maximal domain of validity; rational Szegö-Radau quadrature; rational Szegö-Lobatto quadrature

Citations: Zbl 1109.65027

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