Recovering functions from the spherical mean transform with limited radii data by expansion into spherical harmonics

The aim of the article is to generalize the method presented in [3, Theorem 1] by G. Ambartsoumian, R. Gouia-Zarrad and M. Lewis for recovering functions from their spherical mean transform with limited radii data from the two dimensional case to the general n dimensional case. The idea behind the method is to expand each function in question into spherical harmonics and then obtain, for each term in the expansion, an integral equation of Volterra's type that can be solved iteratively. We show also how this method can be modified for the spherical case of recovering functions from the spherical transform with limited radii data. Lastly, we solve the analogous problem for the case of the Funk transform by again using expansion into spherical harmonics and then obtain an Abel type integral equation which can be inverted by a method introduced in [14].