Convolution on the $n$-Sphere With Application to PDF Modeling

In this paper, we derive an explicit form of the convolution theorem for functions on an $n$ -sphere. Our motivation comes from the design of a probability density estimator for $n$ -dimensional random vectors. We propose a probability density function (pdf) estimation method that uses the derived convolution result on $S^{n}$. Random samples are mapped onto the $n$ -sphere and estimation is performed in the new domain by convolving the samples with the smoothing kernel density. The convolution is carried out in the spectral domain. Samples are mapped between the $n$-sphere and the $n$-dimensional Euclidean space by the generalized stereographic projection. We apply the proposed model to several synthetic and real-world data sets and discuss the results.