Spherical functions on Euclidean space
We study spherical functions on Euclidean spaces from the viewpoint of Riemannian symmetric spaces. Here the Euclidean space E n = G/K where G is the semidirect product R n · K of the translation group with a closed subgroup K of the orthogonal group O(n). We give exact parameterizations of the space of (G, K)—spherical functions by a certain affine algebraic variety, and of the positive definite ones by a real form of that variety. We give exact formulae for the spherical functions in the case where K is transitive on the unit sphere in E n .