  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
