A novel approach through spherical functions in the characterization of invariant functions

Given a compact subgroup K of the orthogonal group acting on the Euclidean space Rn, Gerald Schwarz proved that every smooth K-invariant function on Rn can be expressed as a smooth function of a generating set of $K$-invariant polynomials on n variables. The goal of this work is to provide an alternative and more straightforward proof of this result, based on Gelfand theory, with a particular focus on spherical functions.