Support Theorems for Horocycles on Hyperbolic Spaces

(i) An explicit description of the range D(X)̂where X is a Euclidean space or a symmetric space of the noncompact type ([2], [3]). Here (D = Cc ). In the first case, f̂ in (1.1) is integration over hyperplanes in X = Rn; in the latter case f̂ in (1.1) refers to integration over horocycles ξ in the symmetric space X. (ii) Medical application in X-ray reconstruction ([6], p.47). (iii) Existence theorem for invariant differential equations on a symmetric space X ([3], Lemma 8.1 and Theorem 8.2).