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# Note on Cubature Formulae and Designs Obtained from Group Orbits

Published:2011-11-03
Printed: Dec 2012
• Hiroshi Nozaki,
Graduate School of Information Sciences, Tohoku University, Aoba-ku, Sendai 980-8579, Japan
• Masanori Sawa,
Graduate School of Information Sciences, Nagoya University, Chikusa-ku, Nagoya 464-8601, Japan
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## Abstract

In 1960, Sobolev proved that for a finite reflection group $G$, a $G$-invariant cubature formula is of degree $t$ if and only if it is exact for all $G$-invariant polynomials of degree at most $t$. In this paper, we find some observations on invariant cubature formulas and Euclidean designs in connection with the Sobolev theorem. First, we give an alternative proof of theorems by Xu (1998) on necessary and sufficient conditions for the existence of cubature formulas with some strong symmetry. The new proof is shorter and simpler compared to the original one by Xu, and moreover gives a general interpretation of the analytically-written conditions of Xu's theorems. Second, we extend a theorem by Neumaier and Seidel (1988) on Euclidean designs to invariant Euclidean designs, and thereby classify tight Euclidean designs obtained from unions of the orbits of the corner vectors. This result generalizes a theorem of Bajnok (2007) which classifies tight Euclidean designs invariant under the Weyl group of type $B$ to other finite reflection groups.
 Keywords: cubature formula, Euclidean design, radially symmetric integral, reflection group, Sobolev theorem
 MSC Classifications: 65D32 - Quadrature and cubature formulas 05E99 - None of the above, but in this section 51M99 - None of the above, but in this section