Strict positive definiteness of convolutional and axially symmetric kernels on d-dimensional spheres

Abstract

The paper introduces new sufficient conditions of strict positive definiteness for kernels on d-dimensional spheres which are not radially symmetric but possess specific coefficient structures. The results use the series expansion of the kernel in spherical harmonics. The kernels either have a convolutional form or are axially symmetric with respect to one axis. The given results on convolutional kernels generalise the result derived by Chen et al. [8] for radial kernels. Keywords: strictly positive definite kernels, covariance functions, sphere