Inversion of the fixed radius spherical mean Radon transform of F. John in R 2 and R 3

Following J. Radon’s 1917 celebrated paper, F. John has tried to solve Radon’s problem on the set of all circles of fixed radius a in R 2 . But he rapidly realized that the task was really challenging. Much later returning to this problem in R 3 , he succeeded, using a special solution of a Cauchy problem for the wave equation, to establish an inversion formula under the form of an infinite series of ‘partial’ function reconstructions, each one of them expressible in terms of spherical means of the Radon data on spheres of increasing radius ( 2 m + 1 ) a with m = 0 , 1 , 2 , … . The aim of this paper is to provide a proof of the validity of F. John’s result by reproducing his remarkable inversion formula via a calculation of the formal inversion kernel of this transform based on the polar expansion of 1 / sin ⁡ z . We then show that a Fourier approach to the previous Cauchy problem for the R 3 wave equation is the logical framework for finding F. John’s special wave function as well as for establishing the corresponding inversion scheme. As no solution to this problem in R 2 pre-exists, we move to compute the formal inversion kernel of this transform with the pole expansion of 1 / J 0 ( z ) , where J 0 ( z ) is a Bessel function to obtain a new similar series type inversion formula. Interestingly this result emerges naturally from the Fourier approach of the Darboux equation in R 2 , instead of the wave equation.