Monomial orthogonal polynomials of several variables

A monomial orthogonal polynomial of several variables is of the form xα-Qα(x) for a multiindex α∈N0d+1 and it has the least L2 norm among all polynomials of the form xα-P(x), where P and Qα are polynomials of degree less than the total degree of xα. We study monomial orthogonal polynomials with respect to the weight function ∏i=1d+1|xi|2κi on the unit sphere Sd as well as for the related weight functions on the unit ball and on the standard simplex. The results include explicit formula, L2 norm, and explicit expansion in terms of known orthonormal basis. Furthermore, in the case of κ1=⋯=κd+1, an explicit basis for symmetric orthogonal polynomials is also given.