Note on Cubature Formulae and Designs Obtained from Group Orbits
Printed: Dec 2012
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.
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.
cubature formula, Euclidean design, radially symmetric integral, reflection group, Sobolev theorem
65D32 - Quadrature and cubature formulas
05E99 - None of the above, but in this section
51M99 - None of the above, but in this section