Marginal densities of radially symmetric densities in two and three dimensions

Necessary and sufficient conditions are given for a function p(x) on 0 < absolute value (x) < R to be the marginal density of a radially symmetric density f(r) in the case of two and three dimensions. The first case relies on an integral transform due to Bell (CERN/MPS/DL 73-3), while for 3-space the theory is considerably simpler. The two cases appear to be quite different, and no generalization to n-space is known.