Schwartz Space and Radial Distribution on the Euclidean Motion Group

Let G =R2⋊TbetheEuclidean motion group and let K(λ,t) = I0(λ)δ(t) be a distribution on G, where I0(λ) is the Bessel function of order zero and δ(t) is the Dirac measure on SO(2) ∼ =T, the circle group. In this work, it is proved, among other things, that the distribution K(λ,t) is tempered, positive definite, bounded and radial. Further more, a description of temperature function on G ,realised as the positive definite solution of the Laplace-Beltrami operator on SE(2), is presented.