On two families of Funk-type transforms
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On two families of Funk-type transforms M. Agranovsky1,2 · B. Rubin3 Received: 19 August 2019 / Revised: 19 August 2019 / Accepted: 28 August 2020 © Springer Nature Switzerland AG 2020
Abstract We consider two families of Funk-type transforms that assign to a function on the unit sphere the integrals of that function over spherical sections by planes of fixed dimension. Transforms of the first kind are generated by planes passing through a fixed center outside the sphere. Similar transforms with interior center and with center on the sphere itself we studied in previous publications. Transforms of the second kind, or the parallel slice transforms, correspond to planes that are parallel to a fixed direction. We show that the Funk-type transforms with exterior center express through the parallel slice transforms and the latter are intimately related to the Radon–John dplane transforms on the Euclidean ball. These results allow us to investigate injectivity of our transforms and obtain inversion formulas for them. We also establish connection between the Funk-type transforms of different dimensions with arbitrary center. Keywords Shifted Funk transforms · Radon transforms · Inversion formulas · Moebius transforms Mathematics Subject Classification 44A12 · 37E30
1 Introduction The present investigation deals with a pair of integral operators on the unit sphere Sn in Rn+1 that arise in spherical tomography [8,13] and resemble the classical Funk
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B. Rubin [email protected] M. Agranovsky [email protected]
1
Department of Mathematics, Bar Ilan University, 5290002 Ramat-Gan, Israel
2
Holon Institute of Technology, 5810201 Holon, Israel
3
Department of Mathematics, Louisiana State University, 303 Lockett Hall, Baton Rouge, LA 70803, USA 0123456789().: V,-vol
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M. Agranovsky, B. Rubin
transform. These operators have the form (Fa f )(τ ) =
f (x) dσ (x),
Sn ∩ τ
(Πa f )(ζ ) =
f (x) dσ (x).
(1.1)
Sn ∩ ζ
In the first integral, a function f is integrated over the spherical section Sn ∩ τ by the k-dimensional plane τ passing through a fixed point a ∈ Rn+1 , 1 < k ≤ n. In the second integral, the spherical section Sn ∩ ζ is determined by the k-plane ζ , which is parallel to the vector a = 0. We call these operators the shifted Funk transform and the spherical slice transform, respectively. Our main concern is injectivity of the operators (1.1) and inversion formulas. The case when a = 0 is the origin of Rn+1 corresponds to the totally geodesic transform F0 , which goes back to Minkowski [15,16], Funk [4,5], and Helgason [9]; see also [7,10,21,24] and references therein. This operator annihilates odd functions and can be explicitly inverted on even functions. The case |a| = 1, when Fa is injective, was considered by Abouelaz and Daher [1] (on zonal functions) and Helgason [10, p. 145] (on arbitrary functions) for n = 2. The general case of all n ≥ 2 for hyperplane sections was studied by the second co-author [24, Section 7.2]. The method of this work extends to all 1 < k ≤ n and inversion f
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