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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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
Keywords
- Positive quadrature formulas
- Lobatto–Radau quadrature formulas
- Orthogonal polynomials
- Interlacing property