A Note On Characterizations of Spherical t-Designs

A set $X_N=\{x_1,\dots ,x_N\}$ of $N$ points on the unit sphere ${\mathbb{S}}^d,d\ge 2$ 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 $X_N$. This paper extends characterizations of spherical $t$-designs in previous paper from ${\mathbb{S}}^2$ to general ${\mathbb{S}}^d$. We show that for $N\ge \dim ({\mathbb{P}}_{t+1})$, $X_N$ is a stationary point set of a certain non-negative quantity $A_{N,t}$ and a fundamental system for polynomial space over ${\mathbb{S}}^d$ with degree at most $t$, then $X_N$ is a spherical $t$-design. In contrast, we present that with $N \geq \dim( $\mathbb{P}${t})$,afundamentalsystem$X_N$isaspherical$t$-designifandonlyifnon-negativequantity$D{N,,t}$ vanishes. In addition, the still unanswered questions about construction of spherical $t$-designs are discussed.