We consider the operator,$f$(Δ) for Δ the Laplacian, on spaces of measures on the sphere in$R$^{$d$}, show how to determine a family of approximating kernels for this operator assuming that certain technical conditions are satisfied, and give estimates for the$L$^{2}-norm of$f$(Δ)μ in terms of the energy of the measure μ. We derive a formula, analogous to the classical formula relating the energy of a measure on$R$^{$d$}with its Fourier transform, comparing the energy of a measure on the sphere with the size of its spherical harmonics. An application is given to pluriharmonic measures.