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Gaussian quadrature for sums: A rapidly convergent summation scheme
Author(s):
H.
Monien.
Journal:
Math. Comp.
79
(2010),
857-869.
MSC (2000):
Primary 40A25;
Secondary 33C90, 65D32, 33F05
Posted:
July 21, 2009
MathSciNet review:
2600547
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Additional information
Abstract:
Gaussian quadrature is a well-known technique for numerical integration. Recently Gaussian quadrature with respect to discrete measures corresponding to finite sums has found some new interest. In this paper we apply these ideas to infinite sums in general and give an explicit construction for the weights and abscissae of Gaussian formulas. The abscissae of the Gaussian summation have a very interesting asymptotic distribution function with a kink singularity. We apply the Gaussian summation technique to two problems which have been discussed in the literature. We find that the Gaussian summation has a very rapid convergence rate for the Hardy-Littlewood sum for a large range of parameters.
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Additional Information:
H.
Monien
Affiliation:
Bethe Center for Theoretical Physics, Universität Bonn, Nussallee 12, 53115 Bonn, Germany
Email:
monien@th.physik.uni-bonn.de
DOI:
10.1090/S0025-5718-09-02289-3
PII:
S 0025-5718(09)02289-3
Received by editor(s):
December 19, 2006
Received by editor(s) in revised form:
October 16, 2008 and April 3, 2009
Posted:
July 21, 2009
Copyright of article:
Copyright
2009,
American Mathematical Society
The copyright for this article reverts to public domain after 28 years from publication.
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