On the Determination of a Function from Spherical Averages

Source abstract

Let f be a function $f:\mathbb{R}^{n + 1} \to \mathbb{R}$ which is even in the last variable, i.e., such that $f(x, - y) = f(x,y)$ where $x \in \mathbb{R}^n ,y \in \mathbb{R}$. The mapping R is defined by $f \mapsto Rf = g$ 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.