  We prove that any isotropic positive definite function on the sphere can be
written as the spherical self-convolution of an isotropic real-valued function.
It is known that isotropic positive definite functions on d-dimensional
Euclidean space admit a continuous derivative of order [(d-1)/2]. We show that
the same holds true for isotropic positive definite functions on spheres and
prove that this result is optimal for all odd dimensions.
