Spherical Orthogonal Polynomials and Symbolic-Numeric Gaussian Cubature Formulas

It is well-known that the classical univariate orthogonal polynomials give rise to highly efficient Gaussian quadrature rules. We show how these classical families of polynomials can be generalized to a multivariate setting and how this generalization leads to truly Gaussian cubature rules for specific families of multivariate polynomials. The multivariate homogeneous orthogonal functions that we discuss here satisfy a unique slice projection property: they project to univariate orthogonal polynomials on every one-dimensional subspace spanned by a vector from the unit hypersphere. We therefore call them spherical orthogonal polynomials. 1 Spherical orthogonal polynomials The orthogonal polynomials under discussion were first introduced in [1] in a different form and later in [3] in the current form. Originally they were not termed spherical orthogonal polynomials because of a lack of insight into the mechanism behind the definition. In this paper we give several examples of these families of spherical orthogonal polynomials, present graphical illustrations of the bivariate case, compare them to radial orthogonal polynomials which are a special case of radial basis functions, and discuss some Gaussian cubature formulas which can be derived from the spherical orthogonal polynomials. In dealing with multivariate polynomials and functions we shall often switch between the cartesian and the spherical coordinate system. The cartesian coordinates X = (x1, . . . , xn) ∈ R are then replaced by X = (x1, . . . , xn) = (ξ1z, . . . , ξnz) with ξk, z ∈ R where the directional vector ξ = (ξ1, . . . , ξn) belongs to the unit sphere Sn = {ξ : ||ξ||p = 1}. Here || · ||p denotes one of the usual Minkowski norms. While ξ contains the directional information of X , the variable z contains the (possibly signed) distance information. With the multi-index κ = (κ1, . . . , κn) ∈ N the notations X, κ! and |κ| respectively denote X = x1 1 . . . x κn n κ! = κ1! . . . κn! |κ| = κ1 + . . .+ κn 2 Since z can be positive as well as negative and hence two directional vectors can generate X , we also introduce a signed distance function sd(X) = sgn(x1)||X ||p For the sequel of the discussion we need some more notation. We denote by R[z] the linear space of polynomials in the variable z with real coefficients, by R[ξ] = R[ξ1, . . . , ξn] the linear space of n-variate polynomials in ξk with real coefficients, by R(ξ) = R(ξ1, . . . , ξn) the commutative field of rational functions in ξk and with real coefficients, by R(ξ)[z] the linear space of polynomials in the variable z with coefficients from R(ξ) and by R[ξ][z] the linear space of polynomials in the variable z with coefficients from R[ξ]. Let us introduce the linear functional Γ acting on the variable z, as Γ (z) = ci(ξ) where ci(ξ) is a homogeneous expression of degree i in the ξk: ci(ξ) = X |κ|=i cκξ κ (1) For our purpose cκ = |κ|! κ! Z