On the Determination of a Function from Spherical Averages

Let f be a function f:Rn + 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.