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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.
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Ei. Bannai, Et. Bannai, M. Hirao, M. Sawa. Cubature formulas in numerical analysis and Euclidean tight designs. Europ. J. Combin
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