Orthogonal polynomials and cubature formulae on balls, simplices, and spheres

We report on recent developments on orthogonal polynomials and cubature formulae on the unit ball Bd, the standard simplex Td, and the unit sphere Sd. The main result shows that orthogonal structures and cubature formulae for these three regions are closely related. This provides a way to study the structure of orthogonal polynomials; for example, it allows us to use the theory of h-harmonics to study orthogonal polynomials on Bd and on Td. It also provides a way to construct new cubature formulae on these regions.