Asymptotic Expressions of Several Distributions on the Sphere

Abstract

How to define the products of distributions is a difficult and not completely understood problem, and has been studied from several points of views since Schwartz established the theory of distributions by treating singular functions as linear and continuous functions on the testing function space. Many fields, such as differential equations or quantum mechanics, require such multiplications. In this paper, we use the Temple delta sequence and the convolution given on the regular manifolds to derive an invariant theorem, that powerfully changes the products of distributions of several dimensional spaces into the well-defined products of a single variable. With the help of the invariant theorem, we solve a couple of particular distributional products and hence we are able to obtain asymptotic expressions for $$\displaystyle \delta^{(k)}(\frac{1}{a(r)}(r - t))$$ as well as the distribution $$\displaystyle \delta^{(k)}(\frac{1}{a(r)}(r^2 - t^2))$$ by the Fourier transform, where the distribution $$\delta^{(k)}(r - t)$$ focused on the sphere $$O_t$$ is defined by \[ (\delta^{(k)}(r - t), \, \phi) = \frac{(-1)^k}{t^{n - 1}} \int_{O_t} \frac{\partial ^k}{\partial r^k }(\phi r^{n - 1}) d O_t. \]