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 on associates to a point function the hyperplane function by integration over the hyperplane . If is the dual transform, we can invert 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 -invariant map between representations of the group of isometries on function spaces attached to . 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 acts on each irreducible is given by a simple integral. Thus we obtain an explicit formula for . The action of is immediately related to the spectrum of . This shows that can be inverted by a polynomial in the Laplace-Beltrami operator. Similar procedures give corresponding results for the other compact -point homogeneous spaces: , , , 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