Nonparametric Estimation of a Probability Density on a Riemannian Manifold Using Fourier Expansions

Supposing a given collection y₁, ⋯, yN of i.i.d. random points on a Riemannian manifold, we discuss how to estimate the underlying distribution from a differential geometric viewpoint. The main hypothesis is that the manifold is closed and that the distribution is (sufficiently) smooth. Under such a hypothesis a convergence arbitrarily close to the N-1/2 rate is possible, both in the L₂ and the L_∞ senses.