q-moments remove the degeneracy associated with the inversion of the q-Fourier transform

It was recently proven (Hilhorst 2010 J. Stat. Mech. P10023) that the q-generalization of the Fourier transform is not invertible in the full space of probability density functions for q > 1. It has also been recently shown that this complication disappears if we dispose of the q-Fourier transform not only of the function itself, but also of all of its shifts (Jauregui and Tsallis 2011 Phys. Lett. A 375 2085). Here we show that another route exists for completely removing the degeneracy associated with the inversion of the q-Fourier transform of a given probability density function. Indeed, it is possible to determine this density if we dispose of some extra information related to its q-moments.