The distributional products on spheres and Pizetti's formula

Oa ∂k ∂rk (φrn−1)dσ . Taking the Fourier transform of the distribution and the integral representation of the Bessel function, we obtain asymptotic expansions of δ(k)(r −a) for k = 0, 1, 2, . . . in terms of △j δ(x1, . . . , xn), in order to show the well-known Pizetti formula by a new method. Furthermore, we derive an asymptotic product of φ(x1, . . . , xn) δ(k)(r − a), where φ is an infinitely differentiable function, based on the formula of △m(φψ), and hence we are able to characterize the distributions focused on spheres, which can be written as the sums of multiplet layers in the Gel’fand sense. © 2010 Elsevier B.V. All rights reserved.