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 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 example, 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. (C) 2001 Elsevier Science B.V. All rights reserved.