  To numerically approximate Borel probability measures by finite atomic
measures, we study the spectral decomposition of discrepancy kernels when
restricted to compact subsets of $\mathbb{R}^d$. For restrictions to the
Euclidean ball in odd dimensions, to the rotation group $SO(3)$, and to the
Grassmannian manifold $\mathcal{G}_{2,4}$, we compute the kernels' Fourier
coefficients and determine their asymptotics. The $L_2$-discrepancy is then
expressed in the Fourier domain that enables efficient numerical minimization
based on the nonequispaced fast Fourier transform. For $SO(3)$, the
nonequispaced fast Fourier transform is publicly available, and, for
$\mathcal{G}_{2,4}$, the transform is derived here. We also provide numerical
experiments for $SO(3)$ and $\mathcal{G}_{2,4}$.
