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Spherical harmonics and integral geometry on projective spaces


Author: Eric L. Grinberg
Journal: Trans. Amer. Math. Soc. 279 (1983), 187-203
MSC: Primary 53C65; Secondary 43A90, 58G15
DOI: https://doi.org/10.1090/S0002-9947-1983-0704609-1
MathSciNet review: 704609
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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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Additional Information

DOI: https://doi.org/10.1090/S0002-9947-1983-0704609-1
Keywords: Integral geometry, Radon transform, spherical harmonics, projective spaces
Article copyright: © Copyright 1983 American Mathematical Society

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