Spherical harmonics and integral geometry on projective spaces
Author:
Eric L. Grinberg
Journal:
Trans. Amer. Math. Soc. 279 (1983), 187203
MSC:
Primary 53C65; Secondary 43A90, 58G15
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 LaplaceBeltrami 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 LaplaceBeltrami operator. Similar procedures give corresponding results for the other compact point homogeneous spaces: , , , as well as spheres.
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Additional Information
DOI:
http://dx.doi.org/10.1090/S00029947198307046091
PII:
S 00029947(1983)07046091
Keywords:
Integral geometry,
Radon transform,
spherical harmonics,
projective spaces
Article copyright:
© Copyright 1983
American Mathematical Society
