On positive cubature rules on the simplex and isometric embeddings
Masanori Sawa, Yuan Xu · Mathematics of Computation · 2013
Positive cubature rules of degree 4 4 and 5 5 on the d d -dimensional simplex are constructed for a range of dimensions d d and used to construct cubature rules of index 8 8 or degree 9 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 + ⋯ + x d 2 ) t (x_1^2+ \cdots + x_d^2)^t in terms of linear forms of degree 2 t 2t with rational coefficients for t = 4 t = 4 and 5 5 .