Positive semidefinite kernels that are axially symmetric on the sphere and stationary in time: spectral and semi-spectral theory, and constructive approaches

Positive semidefinite kernels on spheres cross time are on demand in spatial statistics, machine learning and numerical analysis, motivated by applications in the climate, environmental and earth sciences. Most of the recent literature has focused on kernels that are isotropic on the sphere and stationary in time, but this assumption is often overly limited in quantifying spatial dependence properly. A broader class of kernels can be obtained by trading isotropy for axial symmetry, i.e., stationarity with respect to longitudes only. This paper challenges theoretical aspects by providing a spectral representation that uniquely characterizes space-time positive semidefinite kernels that are axially symmetric on the sphere and stationary in time. We also address the problem of partial Fourier inversion and show that it is feasible under mild technical conditions. The second part of the paper focuses on constructive methods to build nonseparable kernels having axial symmetry on the sphere and stationarity over time.