Asymptotic Expressions of Several Distributions on the Sphere

How to define the products of distributions is a difficult and not completely understood problem, andhas been studied from several points of views since Schwartz established the theory of distributionsby 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. Inthis paper, we use the Temple delta sequence and the convolution given on the regular manifoldsto derive an invariant theorem, that powerfully changes the products of distributions of severaldimensional spaces into the well-defined products of a single variable. With the help of the invarianttheorem, we solve a couple of particular distributional products and hence we are able to obtainasymptotic expressions forδ(k)(1a(r)(r−t))as well as the distributionδ(k)(1a(r)(r2−t2))by theFourier transform, where the distributionδ(k)(r−t)focused on the sphereOtis defined by(δ(k)(r−t), φ) =(−1)ktn−1∫Ot∂k∂rk(φrn−1)dOt