On the Degree of Exactness of Some Positive Cubature Formulas on the Sphere

In [3] we studied some interpolatory cubature formulas associated to a fundamental system of (n+1) (n ∈ N odd) points on the sphere, equidistributed on n+1 latitudinal circles. Being interpolatory, these formulas have the degree of exactness n, meaning that they are exact for spherical polynomials of degree ≤ n. We gave also equivalent conditions under which the degree of exactness is n + 1. In this paper we show that n + 1 is the maximal degree of exactness attained by these formulas, under the assumption that the weights are positive. 1 Preliminaries Let S2 = {x ∈ R3 : ‖x‖2 = 1} denote the unit sphere of the Euclidean space R3 and let Ψ : [0, π]× [0, 2π) → S, (ρ, θ) 7→ (sin ρ cos θ, sin ρ sin θ, cos ρ) be its parametrization in spherical coordinates (ρ, θ). The coordinate ρ of a point ξ(Ψ(ρ, θ)) ∈ S2 is usually called the latitude of ξ. We denote by Πn the set of univariate polynomials of degree less than or equal to n, by Pk, k = 0, 1, . . . , the Legendre polynomials of degree k on [−1, 1], normalized by the condition Pk(1) = 1 and by Vn be the space of spherical polynomials of degree less than or equal to n. The dimension of Vn is dimVn = (n+ 1) 2 = N and an orthogonal basis of Vn is given by { Y l m(θ, ρ) = P |l| m (cos ρ)e , −m ≤ l ≤ m, 0 ≤ m ≤ n } . Here P ν m denotes the associated Legendre functions, defined by P ν m(t) = ( (k − ν)! (k + ν)! )1/2 (1− t) d dtν Pm(t), ν = 0, . . . ,m, t ∈ [−1, 1] and for given functions f, g : S2 → C, the inner product is taken as 〈f, g〉 = ∫ S2 f(ξ)g(ξ) dω(ξ), 8 On the Degree of Exactness of Some Positive Cubature Formulas on the Sphere where dω(ξ) stands for the surface element of the sphere. The reproducing kernel of the space Vn is defined by Kn(ξ, η) = n ∑ k=0 2k + 1 4π Pk(ξ ∙ η) = kn(ξ ∙ η), ξ, η ∈ S . For given n we consider a set of points {ξi}i=1,...,N ⊂ S2 and the polynomial functions φi : S 2 → C, i = 1, . . . , N, defined by φi (◦) = Kn(ξi, ◦) = n ∑ k=0 2k + 1 4π Pk(ξi ∙ ◦), i = 1, . . . , N. These polynomials are called scaling functions. A set of points {ξi}i=1,...,N for which the scaling functions {φi }i=1,...,N constitute a basis for Vn is called a fundamental system for Vn. Recently, Láın Fernández proved the following result. Proposition 1.1 [1, 2] Let n ∈ N be an odd number, α ∈ (0, 2) and let 0 < ρ1 < ρ2 < . . . < ρn+1 2 < π/2, ρn+2−j = π− ρj , j = 1, . . . , (n+1)/2, denote a system of symmetric latitudes. Then the set of points Sn(α) = { ξj,k(Ψ(ρj , θ j k)), j, k = 1, . . . , n+ 1 } , where θ j k = { 2kπ n+1 , if j is odd, 2(k−1)+α n+1 π, if j is even, constitutes a fundamental system for Vn. Let us mention that for α = 0 or α = 2, the set Sn(α) does not constitute a fundamental system of points and not many fundamental systems of points are known in the present. In [3] we studied the interpolatory cubature formula ∫ S2 F (ξ) dω(ξ) ≈ n+1 ∑