  By recourse to tempered ultradistributions, we show here that the effect of a
q-Fourier transform (qFT) is to map {\it equivalence classes} of functions into
other classes in a one-to-one fashion. This suggests that Tsallis' q-statistics
may revolve around equivalence classes of distributions and not on individual
ones, as orthodox statistics does. We solve here the qFT's non-invertibility
issue, but discover a problem that remains open.
