## Cubature formulas for symmetric measures in higher dimensions with few points

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- by Aicke Hinrichs and Erich Novak PDF
- Math. Comp.
**76**(2007), 1357-1372 Request permission

## Abstract:

We study cubature formulas for $d$-dimensional integrals with an arbitrary symmetric weight function of product form. We present a construction that yields a high polynomial exactness: for fixed degree $\ell =5$ or $\ell =7$ and large dimension $d$ the number of knots is only slightly larger than the lower bound of Möller and much smaller compared to the known constructions. We also show, for any odd degree $\ell = 2k+1$, that the minimal number of points is almost independent of the weight function. This is also true for the integration over the (Euclidean) sphere.## References

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## Additional Information

**Aicke Hinrichs**- Affiliation: Mathematisches Institut, Universität Jena, Ernst-Abbe-Platz 2, D-07743 Jena, Germany
- Email: hinrichs@math.uni-jena.de
**Erich Novak**- Affiliation: Mathematisches Institut, Universität Jena, Ernst-Abbe-Platz 2, D-07743 Jena, Germany
- Email: novak@math.uni-jena.de
- Received by editor(s): August 25, 2005
- Received by editor(s) in revised form: June 16, 2006
- Published electronically: February 16, 2007
- Additional Notes: Research of the first author was supported by the DFG Emmy-Noether grant Hi 584/2-4.
- © Copyright 2007
American Mathematical Society

The copyright for this article reverts to public domain 28 years after publication. - Journal: Math. Comp.
**76**(2007), 1357-1372 - MSC (2000): Primary 65D32
- DOI: https://doi.org/10.1090/S0025-5718-07-01974-6
- MathSciNet review: 2299778