The bounded spherical functions on the Cartan Motion group and Generalizations for the eigenspaces of the Laplacian on Rn

TL;DR

This paper characterizes bounded spherical functions on real Cartan Motion groups and explores eigenfunctions of the Laplacian on R^n, extending classical harmonic analysis results to more general settings.

Contribution

It generalizes the description of spherical functions for Cartan Motion groups and analyzes Laplacian eigenspaces on R^n with transitive symmetry groups.

Findings

Explicit characterization of bounded spherical functions for real Cartan Motion groups

Extension of Laplacian eigenspace analysis to R^n with transitive group actions

Connections between harmonic analysis on groups and Euclidean spaces

Abstract

The bounded spherical functions are determined for a real Cartan Motion group which is a generalization for the case when the Cartan Motion group is complex written by Helgason Sigurdur . Also, I will do a further step of the Laplacian on R n . I consider the case when K is transitive on the spheres about 0 in Rn ,n > 1