Derivatives of isotropic positive definite functions on spheres

We show that isotropic positive definite functions on the $d$-dimensional sphere which are $2k$ times differentiable at zero have $2k+[(d-1)/2]$ continuous derivatives on $(0,\pi)$. This result is analogous to the result for radial positive definite functions on Euclidean spaces. We prove optimality of the result for all odd dimensions. The proof relies on mont\ée, descente and turning bands operators on spheres which parallel the corresponding operators originating in the work of Matheron for radial positive definite functions on Euclidian spaces.