For centers outside the sphere, the shifted Funk transform reduces, via a Möbius change of variables, to a parallel slice transform, yielding exact injectivity and inversion results.
Non-geodesic Spherical Funk Transforms with One and Two Centers
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abstract
We study non-geodesic Funk-type transforms associated with cross-sections of the n-sphere by k-dimensional planes passing through an arbitrary fixed point inside the sphere. The main results include injectivity conditions for these transforms, inversion formulas, and connection with geodesic Funk transforms. We also show that, unlike the case of planes through a single common center, the integrals over spherical sections by planes through two distinct centers provide the corresponding reconstruction problem a unique solution.
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math.FA 1years
2019 1verdicts
CONDITIONAL 1representative citing papers
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On Two Families of Funk-Type Transforms
For centers outside the sphere, the shifted Funk transform reduces, via a Möbius change of variables, to a parallel slice transform, yielding exact injectivity and inversion results.