Convolution roots and differentiability of isotropic positive definite functions on spheres

Isotropic positive definite functions on spheres are relevant in spatial statistics as well as in approximation theory. Spherical convolution has been used to construct such functions in various contexts. We give a positive answer to the natural question of general this construction is. We prove that any isotropic positive definite function on the sphere has a convolution root, in other words, it 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]. Using the existence of convolution roots, we show that the same result holds true for isotropic positive definite functions on spheres.