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Transactions of the American Mathematical Society

Published by the American Mathematical Society since 1900, Transactions of the American Mathematical Society is devoted to longer research articles in all areas of pure and applied mathematics.

ISSN 1088-6850 (online) ISSN 0002-9947 (print)

The 2020 MCQ for Transactions of the American Mathematical Society is 1.48.

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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


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
  • © Copyright 1983 American Mathematical Society
  • Journal: Trans. Amer. Math. Soc. 279 (1983), 187-203
  • MSC: Primary 53C65; Secondary 43A90, 58G15
  • DOI:
  • MathSciNet review: 704609