A Note On Characterizations of Spherical t-Designs

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A set Xₙ={x₁,,xₙ} of N points on the unit sphere Sᵈ,d≥ 2 is a spherical t-design if the average of any polynomial of degree at most t over the sphere is equal to the average value of the polynomial over Xₙ. This paper extends characterizations of spherical t-designs in previous paper from S² to general Sᵈ. We show that for N≥(Pₜ₊₁), Xₙ is a stationary point set of a certain non-negative quantity Aₙ,ₜ and a fundamental system for polynomial space over Sᵈ with degree at most t, then Xₙ is a spherical t-design. In contrast, we present that with N ≥ (Pₜ), a fundamental system Xₙ is a spherical t-design if and only if non-negative quantity Dₙ,ₜ vanishes. In addition, the still unanswered questions about construction of spherical t-designs are discussed.