To recovering the moments from the spherical mean Radon transform (Q776942)

Given a function \(f\) on \({\mathbb R}^2\), the spherical mean Radon transform considered in this paper maps \(f\) into \({\mathcal M} f\), where \({\mathcal M} f(p,t)\) is the average of \(f\) over the circle with center \(p \in S^1\) and radius \(t > 0\). The authors consider the real Fourier series \[ f_{p,t}(\phi) = \sum_k [a_k(p,t) \cos k\phi + b_k(p,t) \sin k\phi], \] where \(\phi\) is the angular parameter along the circle given by \(p\), \(t\). In particular, \(a_0(p,t) = {\mathcal M} f(p,t)\). If \(a_1\) is also known, the other coefficients \(a_k\), \(b_k\) can be computed one by one by solving a series of ODEs. This can be used to develop an inversion formula, and a method of computing the moments of \(f\) directly from \({\mathcal M} f\). However, there is no discussion of how to find \(a_1\).