Spherical harmonics and integral geometry on projective spaces

Abstract

The Radon transform $R$ on ${\mathbf {C}}{P^{\text {n}}}$ associates to a point function $f(x)$ the hyperplane function $Rf(H)$ by integration over the hyperplane $H$. If ${R^t}$ is the dual transform, we can invert ${R^t}R$ by a polynomial in the Laplace-Beltrami operator, and verify the formula of Helgason [