Asymptotic formulas for the dual Radon transform and applications (Q1079771)

Asymptotic formulas for the behavior at infinity of \({\mathfrak R}^ tg(x)\) are given where \({\mathfrak R}^ t\) is the dual Radon transform and \(g\) is a function in the Schwartz space on \(S^{n-1}\times E^ 1\) (or the Hilbert transform of such a function). The coefficients in the asymptotic expansion involve the moments of \(g\) and hence the polynomial consistency conditions of \textit{S. Helgason} [Acta Math. 113, 153--180 (1965; Zbl 0163.16602)] on the range of the Radon transform. It is shown that \(g={\mathfrak R}f\) for some smooth \(f\) that is \(O(| x|^{-n-m-1})\), \(m\geq -1\), as \(| x| \to \infty\) if and only if \(g\) satisfies the first \(m+1\) conditions of Helgason. Inversion formulas for \({\mathfrak R}^ t\) and a related spherical transform are also given.