Some properties of convolution and spherical analysis on the Euclidean motion group

The Euclidean motion group G = S E(n) is a non-commutative and non-compact solvable Lie group which may also be referred to as an affine group of rigid motions on Rn realized as G = Rn x SO(n). It is observed that this group is isomorphic to the collection of homogeneous transformation matrices. This article presents a novel method for calculating the Haar measure of G, and is demonstrated that it is the product of the Lebesque measure of R2 and the normalized measure of SO(2). A thorough description is given of the topology of the Schwartz space of G (S(G = S E(n))) and how it is produced by a system of semi-norms. For n = 2, the convolution product of functions in SS E(2) realises the Radon transform of functions in SS E(2). It is demonstrated that this convolution is homogeneous, smooth and satisfies the moment condition. Further more, spherical analysis on the Gelfand pair (Rn x SO(n), SO(n)) is presented, including an explicit determination of spherical function for G, when n = 2, by the method of separation of variables.