ABSTRACT
The spectral methods are developed based on multivariate polynomial approximations over the unit ball. This chapter is an introduction to such polynomials, especially the use of orthonormal polynomials as a basis for the space of polynomials. It summarizes results on approximation by multivariate polynomials over the unit ball, the rapid evaluation of such polynomials, and numerical quadrature over the unit ball. Univariate orthogonal polynomials satisfy a triple recursion relation for the Gegenbauer polynomials. Such a relation must be created in a more careful way for multivariate orthogonal polynomials. The Clenshaw algorithm for univariate polynomials is an efficient way to evaluate sums when the polynomials satisfy a triple recursion relation.
