To numerically approximate Borel probability measures by finite atomic measures, we study the spectral decomposition of discrepancy kernels when restricted to compact subsets of\documentclass[12pt]{minimal}
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				\begin{document}$$\mathbb {R}^d$$\end{document}Rd. For restrictions to the Euclidean ball in odd dimensions, to the rotation group\documentclass[12pt]{minimal}
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				\begin{document}$$\textrm{SO}(3)$$\end{document}SO(3), and to the Grassmannian manifold\documentclass[12pt]{minimal}
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				\begin{document}$$\mathcal {G}_{2,4}$$\end{document}G2,4, we compute the kernels’ Fourier coefficients and determine their asymptotics. The\documentclass[12pt]{minimal}
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				\begin{document}$$L_2$$\end{document}L2-discrepancy is then expressed in the Fourier domain that enables efficient numerical minimization based on the nonequispaced fast Fourier transform. For\documentclass[12pt]{minimal}
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				\begin{document}$$\textrm{SO}(3)$$\end{document}SO(3), the nonequispaced fast Fourier transform is publicly available, and, for\documentclass[12pt]{minimal}
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				\begin{document}$$\mathcal {G}_{2,4}$$\end{document}G2,4, the transform is derived here. We also provide numerical experiments for\documentclass[12pt]{minimal}
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				\begin{document}$$\textrm{SO}(3)$$\end{document}SO(3)and\documentclass[12pt]{minimal}
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				\begin{document}$$\mathcal {G}_{2,4}$$\end{document}G2,4.