Spherical harmonics and integral geometry on projective spaces

Grinberg, Eric L.

doi:10.1090/S0002-9947-1983-0704609-1

Spherical harmonics and integral geometry on projective spaces
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by Eric L. Grinberg PDF

Trans. Amer. Math. Soc. 279 (1983), 187-203 Request permission

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 [7] 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 ${\mathbf {C}}{P^n}$. Pulling back to a sphere via a 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 ${R^t}R$ is immediately related to the spectrum of ${\mathbf {C}}{P^n}$. This shows that ${R^t}R$ can be inverted by a polynomial in the Laplace-Beltrami operator. Similar procedures give corresponding results for the other compact $2$-point homogeneous spaces: ${\mathbf {R}}{P^n}$, ${\mathbf {H}}{P^n}$, ${\mathbf {O}}{P^n}$, as well as spheres.

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