Convolution roots and differentiability of isotropic positive definite   functions on spheres
  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. 
Johanna Ziegel
math.PR math.ST stat.TH