Unitarization and Inversion Formulae for the Radon Transform Between Dual Pairs

Abstract

We consider the Radon transform associated to dual pairs (X, Ξ) in the sense of Helgason, with X = G/K and Ξ = G/H, where G = R d ⋊ K, K is a closed subgroup of GL(d, R) and H is a closed subgroup of G. Under some technical assumptions, we prove that if the quasi regular representations of G acting on L 2 (X) and L 2 (Ξ) are irreducible, then the Radon transform admits a unitarization intertwining the two representations. If, in addition, the representations are square integrable, we provide an inversion formula for the Radon transform based on the voice transform associated to these representations.

Radon Transforms in Hyperbolic Spaces and Their Discrete Counterparts

“…The series in the above statement is absolutely convergent in the convolution operator norm on 2 . An explicit inversion formula for the X-ray transform can be derived from this fact, see [1,6].…”

“…For the relevant background, the reader is referred to [18]. By Corollary 5.3 we have φ 1/2+it ∈ 2+ , and, by the asymptotic rate of decay of spherical functions, the product φ σ φ 1/2+it is positive definite and belongs to 2 assume z > 1/2. As before, write y z (e) = q −z e .…”

“…Here we briefly explain the setup of the disk U . Denote by dμ the Haar measure on the unimodular group G = Aut + U and by dv the quotient measure on G/G 0 , that is, the volume element on U , namely dv = sinh r dθ dr (2) Let G 0 be the isotropy subgroup at 0 and g → g = gG 0 the canonical projection onto G/G 0 . Via this projection, a function f on U = G/G 0 can be bijectively lifted to a right-G 0 -invariant function on G; the function f is radial around 0 if and only if the lifted function is bi-G 0 -invariant.…”

“…its spectral radius ρ(η 1 ) is (q −1)/2q +1/ √ q, and its resolvent ξ γ (z) at the eigenvalue γ (z) is the convolution operator on 2 given by the function r z (e) = 2q…”

“…Many different variants of the Radon transform have been investigated since then, in a long list of articles in pure and in applied mathematics. Despite its long history, the Radon transform is still in the focus of current research, both in continuous and discrete settings: see, for instance, [2,10,16,34] and their references lists.…”