Strictly positive definite kernels on the $2$-sphere: beyond radial symmetry

Abstract: The paper introduces a new characterisation of strictly positive definiteness for kernels on the 2-sphere without assuming the kernel to be radially (isotropic) or axially symmetric. The results use the series expansion of the kernel in spherical harmonics. Then additional sufficient conditions are proven for kernels with a block structure of expansion coefficients. These generalise the result derived by Chen et al. 2003 for radial kernels to non-radial kernels.