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On domain C (Rf) we invert the Radon transform that maps a function to its mean values on spheres containing the origin. Our inversion formula implies that if f E C '(Rf) and its transform is zero on spheres inside a disc centered at 0, then f is zero inside that disc. We give functions f a C '(R f) whose transforms are identically zero and we give a necessary condition for a function to be the transform of a rapidly decreasing function. We show that every entire function is the transform of a real analytic function. These results imply that smooth solutions to the classical Darboux equation are determined by the data on any characteristic cone with vertex on the initial surface; if the data is zero near the vertex then so is the solution. If the data is entire then a real analytic solution with that data exists. In 1917 Radon inverted the first "Radon transform" [18]. This transform, R, maps a function on Rn to a function on the set of hyperplanes in Rn. If f is a continuous function of compact support on R" then Rf evaluated on a hyperplane is the integral of f over that hyperplane in its natural measure. The case n = 2 has many applications in science, engineering, and medicine [2], [3], [15], [21] and the transform on Rn (n arbitrary) has many applications to partial differential equations [13], [14]. Generalizations of this Radon transform to integrations over certain spheres and ellipsoids have been studied by John and others [13], [19] again in connection with partial differential equations. Moreover these examples are all special cases of the generalized Radon transform: given smooth manifolds X, Y, and a class of submanifolds of X, { Hy I y E Y), one specifies smooth measures on each Hy. The generalized Radon transform R from Co'(X) to functions on Y takes f E C0?(X) to the integrals of f over the manifolds Hy in the measures /, [7]. In many cases restrictions on the support of Rf imply restrictions on the support of f [10]; this fact is useful in applications to partial differential equations [11], [14]. In this article we define a Radon transform over spheres passing through the origin in Rn. 1ff E C(Rn), the transform f evaluated on a sphere containing 0 is the mean value of f over that sphere in its natural measure. Our main result, Theorem 1, is an inversion formula for this transform: if f E C (R n) then f(x) is determined by the values of f on spheres that lie inside the disc of radius Ixi Received by the editors August 17, 1979; presented to the Society, October 19, 1979 at Howard University. AMS (MOS) subject classifications (1970). Primary 44A05, 35Q05.

A Radon transform on spheres through the origin in 𝑅ⁿ and applications to the Darboux equation