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Gaussian, Lobatto and Radau positive quadrature rules with a prescribed abscissa

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Abstract

For a given \(\theta \in (a,b)\), we investigate the question whether there exists a positive quadrature formula with maximal degree of precision which has the prescribed abscissa \(\theta \) plus possibly \(a\) and/or \(b\), the endpoints of the interval of integration. This study relies on recent results on the location of roots of quasi-orthogonal polynomials. The above positive quadrature formulae are useful in studying problems in one-sided polynomial \(L_1\) approximation.

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Acknowledgments

A first version of this paper considered the special case of Jacobi weights on \((a,b)=(-1,1)\). We thank Prof. K. Jordaan who kindly did let us know about her paper [4].

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Authors and Affiliations

  1. Laboratoire Painlevé UMR 8524 (ANO-EDP), UFR Mathématiques, M3, UST Lille, 59655 , Villeneuve d’Ascq Cedex, France

    Bernhard Beckermann

  2. Facultad de Ciencias Físico-Matemáticas (FCFM), Benemerita Universidad Autonoma de Puebla (BUAP), Apartado Postal 1152, Pue. C.P. 72000 , Puebla, México

    Jorge Bustamante & Reinaldo Martínez-Cruz

  3. Departamento de Matemáticas, Universidad de Jaén, Jaén, Spain

    José M. Quesada

Authors
  1. Bernhard Beckermann
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  2. Jorge Bustamante
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  3. Reinaldo Martínez-Cruz
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  4. José M. Quesada
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Corresponding author

Correspondence to Jorge Bustamante.

Additional information

Bernhard Beckermann was supported in part by the Labex CEMPI (ANR-11-LABX-0007-01). José M. Quesada was partially supported by Junta de Andalucía. Research Group FQM0268.

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Beckermann, B., Bustamante, J., Martínez-Cruz, R. et al. Gaussian, Lobatto and Radau positive quadrature rules with a prescribed abscissa. Calcolo 51, 319–328 (2014). https://doi.org/10.1007/s10092-013-0087-3

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  • DOI: https://doi.org/10.1007/s10092-013-0087-3

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