DIFFERENTIABLE POSITIVE DEFINITE KERNELS ON SPHERES

Abstract. We analyze term-by-term differentiability of uniformly con-vergent series of the form k=0 ρkYk(x)Yk(y), x, y ∈ Sm−1, where Sm−1 is the unit sphere in Rm, ρk ≥ 0, k = 0, 1,..., k=0 ρk> 0, and {Yk} is a sequence of spherical harmonics or even more general functions. Since this class of kernels includes the continuous positive definite kernels on Sm−1, the results in this paper will show that, under certain condi-tions, the action of convenient differential operators on positive definite (strictly positive definite) kernels on Sm−1 generate positive definite kernels. 1