By Karl Albrecht

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Example 1. Let χ be the indicator function of a domain D ⊂ S and g be a smooth function in S. The Funk transform of f = gχ is equal to the integral M [f ] (ξ) = gdS D∩ξ ⊥ The wave front of f is contained in the cotangent bundle N ∗ (Γ) of the boundary, moreover it coincides with this bundle if g is nowhere flat in Γ. 7 Transformation of boundary discontinuities . Let D be a domain in En with the smooth boundary Γ = ∂D. Take a smooth function φ in En that is negative in D and positive in the complement and dφ = 0 in Γ; set φ± = max (±φ, 0) .

Let E be an Euclidean space and L be a projective transformation ¯ of E. It defines the mapping L : E\L−1 (H∞ ) → of the projective closure E ¯ This mapping is E\L (H∞ ) where H∞ denotes the improper hyperplane in E. factorable for the variety of k-dimensional affine subspaces A ⊂ E and arbitrary k. , xn ) . e. , xn /x0 ) . Set p0 (x) = x0 . , bk . , ik . , ak . e. 3) t i ai , ti = 1} . , ak by another set of vectors that span A. e. for the case 0 a0 = x, ai = x + εui , p0 u1 = ... = p0 uk = 0.

If α is an integer or inverse integer we can apply an alternative method by reduction to the Radon transform. Let α = 1/n; consider the conformal mapping w = z n defined in E\ {0} . Any curve A1/n coincides with the image of a straight lines L and f (w) ds = f (z) n |z|n−1 ds. L A1/n Form this equation we know all line integrals of g (z) = f (z) n |z|n−1 and can recover g by inverting the plane Radon transform. In the case α = n the image of A2n is again a straight line L and f n |z|n−1 f (z) ds = An −1 ds.

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