On the Determination of a Function from Spherical Averages

Let f be a function f: R^n + 1 R which is even in the last variable, i. e. , such that f (x, - y) = f (x, y) where x Rⁿ, y R. The mapping R is defined by f Rf = g where g (x, r) is the average of f over a sphere with radius r and center at a point (x, 0) in the hyperplane y = 0. The problem to invert the mapping R is studied. Extending the domain of the mapping R to the class of tempered distributions, we give a characterization of the range of R and prove that the inverse mapping R^ - 1 exists and is continuous in the topology of distributions. An inversion formula, first discovered by J. Fawcett, is obtained in terms of Fourier transforms and a Sobolev estimate for the inverse mapping is given. Next, inversion methods using only values of g on some bounded set are studied. First a uniqueness theorem of Courant and Hilbert is generalized to distributions. Inversion formulas involving partial Fourier transforms are given and a numerical inversion procedure is proposed.