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To numerically approximate Borel probability measures by finite atomicmeasures, we study the spectral decomposition of discrepancy kernels whenrestricted to compact subsets of $\mathbb{R}^d$. For restrictions to theEuclidean ball in odd dimensions, to the rotation group $SO(3)$, and to theGrassmannian manifold $\mathcal{G}_{2,4}$, we compute the kernels' Fouriercoefficients and determine their asymptotics. The $L_2$-discrepancy is thenexpressed in the Fourier domain that enables efficient numerical minimizationbased on the nonequispaced fast Fourier transform. For $SO(3)$, thenonequispaced fast Fourier transform is publicly available, and, for$\mathcal{G}_{2,4}$, the transform is derived here. We also provide numericalexperiments for $SO(3)$ and $\mathcal{G}_{2,4}$.

Spectral decomposition of discrepancy kernels on the Euclidean ball, the special orthogonal group, and the Grassmannian manifold