The aim of this article is to derive a reconstruction formula for the recovery of $$C^{1}$$ C 1 functions, defined on the unit sphere $${{\mathbb {S}}}^{n - 1}$$ S n - 1 , given their integrals on a special family of $$n - 2$$ n - 2 dimensional sub-spheres. For a fixed point $$\overline{a}$$ a ¯ strictly inside $${{\mathbb {S}}}^{n - 1}$$ S n - 1 , each sub-sphere in this special family is obtained by intersection of $${{\mathbb {S}}}^{n - 1}$$ S n - 1 with a hyperplane passing through $$\overline{a}$$ a ¯ . The case $$\overline{a} = 0$$ a ¯ = 0 results in an inversion formula for the special case of integration on great spheres (i.e., Funk transform). The limiting case where $$p\in {{\mathbb {S}}}^{n - 1}$$ p ∈ S n - 1 and $$\overline{a}\rightarrow p$$ a ¯ → p results in an inversion formula for the special case of integration on spheres passing through a common point in $${{\mathbb {S}}}^{n - 1}$$ S n - 1 .
Analysis and Mathematical Physics – Springer Journals
Published: May 21, 2016
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