On positive cubature rules on the simplex and isometric embeddings

Sawa, Masanori; Xu, Yuan

doi:10.1090/S0025-5718-2013-02762-7

On positive cubature rules on the simplex and isometric embeddings
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by Masanori Sawa and Yuan Xu PDF

Math. Comp. 83 (2014), 1251-1277 Request permission

Abstract:

Positive cubature rules of degree $4$ and $5$ on the $d$-dimensional simplex are constructed for a range of dimensions $d$ and used to construct cubature rules of index $8$ or degree $9$ on the unit sphere. The latter ones lead to explicit isometric embedding among the classical Banach spaces. Among other things, our results include several explicit representations of $(x_1^2+ \cdots + x_d^2)^t$ in terms of linear forms of degree $2t$ with rational coefficients for $t = 4$ and $5$.

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