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Differentiable positive definite kernels on spheres

Menegatto V. A., Oliveira C. P., Peron A. P.

Journal of Applied Analysis

2009

Vol. 15, nr 1

101-117

We analyze term-by-term differentiability of uniformly convergent series of the form [wzór], where Sm-1 is the unit sphere in Rm, pk ≥ 0, k = 0,1,..., [wzór] Pk > 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 conditions, the action of convenient differential operators on positive definite (strictly positive definite) kernels on Sm-1 generate positive definite kernels.