Spherical Harmonics and Integral Geometry on Projective Spaces
Transactions of the American Mathematical Society
The Radon transform R on CP" associates to a point function/(jc) the hyperplane function Rf(H) by integration over the hyperplane H. Il R' is the dual transform, we can invert R'R by a polynomial in the Laplace-Beltrami operator, and verify the formula of Helgason  with very simple computations. We view the Radon transform as a G-invariant map between representations of the group of isometries G = U(n + 1) on function spaces attached to CP". Pulling back to a sphere via a suitable Hopf
... suitable Hopf fibration and using the theory of spherical harmonics, we can decompose these representations into irreducibles. The scalar by which R acts on each irreducible is given by a simple integral. Thus we obtain an explicit formula for R. The action of R'R is immediately related to the spectrum of CP". This shows that R'R can be inverted by a polynomial in the Laplace-Beltrami operator. Similar procedures give corresponding results for the other compact 2-point homogeneous spaces: RT"1, HP", OP", as well as spheres.