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Positive interpolatory quadrature rules generated by some biorthogonal polynomials


Authors: D. S. Lubinsky and A. Sidi
Journal: Math. Comp. 79 (2010), 845-855
MSC (2000): Primary 41A55, 65D30, 65B99, 42C99
Published electronically: September 18, 2009
MathSciNet review: 2600546
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Abstract: Interpolatory quadrature rules whose abscissas are zeros of a biorthogonal polynomial have proved to be useful, especially in numerical integration of singular integrands. However, the positivity of their weights has remained an open question, in some cases, since 1980. We present a general criterion for this positivity. As a consequence, we establish positivity of the weights in a quadrature rule introduced by the second author in 1980, generated by a polynomial that is biorthogonal to $ \left( \log x\right) ^{j}$, $ 0\leq j\leq n-1$.


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Additional Information

D. S. Lubinsky
Affiliation: School of Mathematics, Georgia Institute of Technology, Atlanta, Georgia 30332-0160
Email: lubinsky@math.gatech.edu

A. Sidi
Affiliation: Department of Computer Science, Technion-Israel Institute of Technology, Haifa 32000 Israel
Email: asidi@cs.technion.ac.il

DOI: https://doi.org/10.1090/S0025-5718-09-02299-6
Keywords: Biorthogonal polynomials, interpolatory quadrature rules, positive weights
Received by editor(s): October 7, 2008
Published electronically: September 18, 2009
Additional Notes: Research supported by NSF grant DMS0400446 and US-Israel BSF grant 2004353.
Article copyright: © Copyright 2009 American Mathematical Society