Marcinkiewicz-Zygmund inequalities for hemispherical $t$-designs

In this paper, we introduce a new set of ${\mathbb{L}}_2$-orthogonal polynomials on the hemisphere ${\mathbb{S}}_{+}^d:=\{\mathbf{x}\in {\mathbb{R}}^{d+1}:\mathrm{\parallel }\mathbf{x}\mathrm{\parallel }=1,\mathbf{x}\cdot {\mathbf{e}}_{d+1}\ge 0\}$ and establish ${\mathbb{L}}_1$ Marcinkiewicz-Zygmund inequalities on the hemisphere for even and odd spherical polynomials. Hemispherical $t$-designs provide equal weight cubature rules on the hemisphere which are exact for polynomials up to degree $t$. We propose variational characterizations of hemispherical $t$-designs by the ${\mathbb{L}}_2$-orthogonal polynomials. Moreover, based on the Marcinkiewicz-Zygmund inequalities and the ${\mathbb{L}}_2$-orthogonal polynomials, we prove that for each $n\ge C_d{t}^d$, there exists a hemispherical $t$-design on the hemisphere with $n$ points, where $C_d$ is a constant depending only on $d$.