Reconstruction of functions on the sphere from their integrals over hyperplane sections

Abstract

We obtain new inversion formulas for the Funk type transforms of two kinds associated to spherical sections by hyperplanes passing through a common point A which lies inside the n-dimensional unit sphere or on the sphere itself. Transforms of the first kind are defined by integration over complete subspheres and can be reduced to the classical Funk transform. Transforms of the second kind perform integration over truncated subspheres, like spherical caps or bowls, and can be reduced to the hyperplane Radon transform. The main tools are analytic families of λ-cosine transforms, Semyanisyi's integrals, and modified stereorgraphic projection with the pole at A. Assumptions for functions are close to minimal.2010 Mathematics Subject Classification. Primary 44A12; Secondary 44A15. Key words and phrases. Radon transform and Funk transform and cosine transform and Semyanistyi integrals and inversion formulas.

Support theorems for Funk-type isodistant Radon transforms on constant curvature spaces

“…Combining the intertwining relations (5.4) for different values of p but with the same curvature leads to intertwining relations similar in spirit to those that are in [3,4,6,7,20,26,30,33,34,38] for the sphere, but also for the hyperbolic case. For instance, we show here two relevant applications of this idea for the sphere S n = K n 1 .…”

“…We start considering the kernels in the elliptic case. This makes a direct generalization of Funk's result [13] and leads to kernel descriptions different than the ones in [7,15,16,20,26,27,30,31,33,34]. Figure 4 shows what is at stake.…”

“…Notice that ⟨ks1⟩ states the injectivity of the spherical slice transform under a mild condition, while the next statements describes the kernels of the spherical shifted Funk transform by a kind of parity condition in accordance with the results of [3,4,6,7,20,26,30,33,34,38]. Since ⟨ks2⟩ served as a prototype for the next statements of the theorem, it is certainly not the sharpest possible version, so we return to it in Sect.…”

“…If C is a ray with fixed p ∈ ℝ ∪ {±∞} , then we call the associated restrictions of the slice transform p-shifted Funk transform 2 and denote it by p . For the sphere ( = 1 ) this was recently quite intensively investigated [3,4,6,7,18,20,25,26,30,[32][33][34]38], but there are also sporadic earlier results [1,16] as well. Surprisingly enough there seems to be no general results for = 0, −1.…”

On Two Families of Funk-Type Transforms

“…Operators F a with |a| < 1 were studied by Salman [26,27] (in a different setting, caused by support restrictions) and later by Quellmalz [17,18] and Rubin [23] for k = n. The case of all 1 < k ≤ n was considered by Agranovsky and Rubin [2]. The kernel of such operators consists of functions f that are odd in a certain sense, whereas the corresponding even components of f can be explicitly reconstructed from F a f .…”