  This work provides theoretical foundations for kernel methods in the
hyperspherical context. Specifically, we characterise the native spaces
(reproducing kernel Hilbert spaces) and the Sobolev spaces associated with
kernels defined over hyperspheres. Our results have direct consequences for
kernel cubature, determining the rate of convergence of the worst case error,
and expanding the applicability of cubature algorithms based on Stein's method.
We first introduce a suitable characterisation on Sobolev spaces on the
$d$-dimensional hypersphere embedded in $(d+1)$-dimensional Euclidean spaces.
Our characterisation is based on the Fourier--Schoenberg sequences associated
with a given kernel. Such sequences are hard (if not impossible) to compute
analytically on $d$-dimensional spheres, but often feasible over Hilbert
spheres. We circumvent this problem by finding a projection operator that
allows to Fourier mapping from Hilbert into finite dimensional hyperspheres. We
illustrate our findings through some parametric families of kernels.
