Spherical transform

Summary

The spherical transform of a Gelfand pair $(G, K)$ integrates a bi-$K$-invariant function against the bounded spherical functions. It is the Gelfand transform of the commutative convolution algebra $L^1(K \backslash G / K)$, so it turns convolution into pointwise multiplication. For $G = \R^n$ with $K$ trivial it is the ordinary Fourier transform, and for $G = \R^n \rtimes \mathrm{SO}(n)$ with $K = \mathrm{SO}(n)$ it is the Fourier transform of radial functions. For semisimple Lie groups its inversion and Plancherel formula are governed by the Harish-Chandra c-function.