On positive cubature rules on the simplex and isometric embeddings

TL;DR

This paper constructs positive cubature rules on the simplex of degree 4 and 5, and uses them to develop cubature rules on the sphere, leading to explicit isometric embeddings among classical Banach spaces.

Contribution

It introduces new positive cubature rules on the simplex and applies them to derive explicit isometric embeddings between Banach spaces.

Findings

Constructed positive cubature rules of degree 4 and 5 on the simplex.

Developed cubature rules of index 8 or degree 9 on the unit sphere.

Provided explicit rational representations of polynomial forms for t=4 and 5.

Abstract

Positive cubature rules of degree 4 and 5 on the $d$-dimensional simplex are constructed 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 $(x_1^2+...+ x_d^2)^t$ in terms of linear forms of degree $2t$ with rational coefficients for t=4 and 5.