Radon transform on spaces of constant curvature

Berenstein, Carlos A.; Tarabusi, Enrico Casadio; Kurusa, Árpád

doi:10.1090/s0002-9939-97-03570-3

Cited by 16 publications

(10 citation statements)

References 14 publications

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“…The case j = 0 corresponds to the totally geodesic transform R k : h → ζ h that takes functions on H n to functions on G H (n, k); cf. [1,2,20,21,26,35,50]. The corresponding dual transform is defined by…”

Section: Radon Transforms Of Zonal Functionsmentioning

confidence: 99%

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Higher-Rank Radon Transforms on Constant Curvature Spaces

Rubin¹

2021

Preprint

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We study higher-rank Radon transforms of the form f (τ ) → τ ⊂ζ f (τ ), where τ is a j-dimensional totally geodesic submanifold in the n-dimensional real constant curvature space and ζ is a similar submanifold of dimension k > j. The corresponding dual transforms are also considered. The transforms are explored the Euclidean case (affine Grassmannian bundles), the elliptic case (compact Grassmannians), and the hyperbolic case (the hyperboloid model, the Beltrami-Klein model, and the projective model). The main objectives are sharp conditions for the existence and injectivity of the Radon transforms in Lebesgue spaces, transition from one model to another, support theorems, and inversion formulas. Conjectures and open problems are discussed.

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Section: Radon Transforms Of Zonal Functionsmentioning

confidence: 99%

“…The corresponding formulas for totally geodesic submanifolds and the relevant Radon transforms require substantial technical work related to computation of Jacobians; cf. [2,26], [39, pp. 412-416].…”

Section: Introductionmentioning

confidence: 99%

Higher-Rank Radon Transforms on Constant Curvature Spaces

Rubin¹

2021

Preprint

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“…Theorems of the previous section can be converted into the similar statements for the Funk type transforms on the sphere and Grassmann manifolds by making use of the stereographic projection. A remarkable interrelation between diverse integral operators on R n and S n is known for many years and the corresponding transition formulas can be found in numerous publications; see, e.g., Mikhlin [45,, Berenstein, Casadio Tarabusi and Kurusa [4], Drury [23], Rubin [56, Section 5.2], to mention a few. Below we recall the reasoning from our paper [54] which has proved to be especially helpful in the general context of Grassmannians.…”

Section: More Generally Assumingmentioning

confidence: 99%

Norm estimates for k-plane transforms and geometric inequalities

Rubin

2019

Advances in Mathematics

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The article is devoted to remarkable interrelation between the norm estimates for k-plane transforms in weighted and unweighted L p spaces and geometric integral inequalities for crosssections of measurable sets in R n . We also consider more general j-plane to k-plane transforms on affine Grassmannians and their compact modifications. The article contains a series of new integral-geometric inequalities with sharp constants, explicit equalities, conjectures, and open problems.

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“…Using the projective equivalence of the operator (7.2) and the hyperplane Radon transform R, as in [22,Lemma 8.3] (see also [13,3]), we obtain the following kernel description, which is implied by Theorem 3.7. We write x ∈ H n in the hyperbolic polar coordinates as x = θ sinhr + e n+1 coshr, θ ∈ S n−1 , r > 0, and consider the Fourier-Laplace coefficients f m,µ (r) =…”

Section: The Totally Geodesic Radon Transform On the Hyperbolic Spacementioning

confidence: 99%

Null spaces of Radon transforms

Estrada

Rubin

2016

Advances in Mathematics

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Abstract. We obtain new descriptions of the null spaces of several projectively equivalent transforms in integral geometry. The paper deals with the hyperplane Radon transform, the totally geodesic transforms on the sphere and the hyperbolic space, the spherical slice transform, and the Cormack-Quinto spherical mean transform for spheres through the origin. The consideration extends to the corresponding dual transforms and the relevant exterior/interior modifications. The method relies on new results for the Gegenbauer-Chebyshev integrals, which generalize Abel type fractional integrals on the positive half-line.

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Radon transform on spaces of constant curvature

Abstract: Abstract. A correspondence among the totally geodesic Radon transformsas well as among their duals-on the constant curvature spaces is established, and is used here to obtain various range characterizations.

Cited by 16 publications

References 14 publications

Higher-Rank Radon Transforms on Constant Curvature Spaces

Higher-Rank Radon Transforms on Constant Curvature Spaces

Norm estimates for k-plane transforms and geometric inequalities

Null spaces of Radon transforms

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