Orthogonal polynomials and cubature formulae on spheres and on simplices

Orthogonal polynomials on the standard simplex E in R are shown to be related to the spherical orthogonal polynomials on the unit sphere S in R that are invariant under the group Z2 x • • • x Z2. For a large class of measures on S, cubature formulae invariant under Z2 X • • • X Z2 are shown to be characterized by cubature formulae on S. Moreover, it also is shown that there is a correspondence between orthogonal polynomials and cubature formulae on S and those invariant on the unit ball B in R. The results provide a new approach to study orthogonal polynomials and cubature formulae on spheres and on simplices.

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