A Duality in Integral Geometry. Generalized Radon Transforms and Orbital Integrals

Abstract

The inversion formulas in Theorems 3.1, 3.3, 3.4 and 3.7, Ch. I suggest the general problem of determining a function on a manifold by means of its integrals over certain submanifolds. In order to provide a natural framework for such problems we consider the Radon transform f → f̂ on ℝⁿ and its dual ϕ → ϕ̌ from a group-theoretic point of view, motivated by the fact that the isometry group M(n) acts transitively on both ℝⁿ and on the hyperplane space ℙⁿ. Thus (1)

$$ {\mathbb{R}^n} = M(n)/0(n){\kern 1pt} ,{\kern 1pt} {\mathbb{P}^n} = M(n)/{\mathbb{Z}_2} \times M(n - 1) $$

((1))

where 𝕆(n) is the orthogonal group fixing the origin 0 ϵ ℝⁿ and Z₂ × M(n-1) is the subgroup of M(n) leaving a certain hyperplane ξ₀ through 0 stable. (Z₂ consists of the identity and the reflection in this hyperplane.)