Spectral Decomposition of Discrepancy Kernels on the Euclidean Ball, the Special Orthogonal Group, and the Grassmannian Manifold

To numerically approximate Borel probability measures by finite atomic measures, we study the spectral decomposition of discrepancy kernels when restricted to compact subsets of [Formula: see text] . For restrictions to the Euclidean ball in odd dimensions, to the rotation group [Formula: see text] , and to the Grassmannian manifold [Formula: see text] , we compute the kernels’ Fourier coefficients and determine their asymptotics. The [Formula: see text] -discrepancy is then expressed in the Fourier domain that enables efficient numerical minimization based on the nonequispaced fast Fourier transform. For [Formula: see text] , the nonequispaced fast Fourier transform is publicly available, and, for [Formula: see text] , the transform is derived here. We also provide numerical experiments for [Formula: see text] and [Formula: see text] .