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

摘要

We report on recent developments on orthogonal polynomials and cubature formulae on the unit ball B~d, the standard simplex T~d, and the unit sphere S~d. 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 examples, it allows us to use the theory of h-harmonics to study orthogonal polynomials on B~d and on T~d. It also provides a way to construct new cubature formulae on these regions.