Christian Hainzl, Robert Seiringer General decomposition of radial functions on R^n and refined conditions for positive definiteness (14K, LaTeX2e) ABSTRACT. We present a generalization of the Fefferman-de la Llave decomposition of the Coulomb potential to quite arbitrary radial functions $V$ on $\R^n$ going to zero at infinity. As a byproduct, we obtain conditions for positive definiteness of $V$, thereby improving results of Askey.