  In the paper the question - Is the q-Fourier transform of a q-Gaussian a
q'-Gaussian (with some q') up to a constant factor? - is studied for the whole
range of $q\in (-\infty, 3)$. This question is connected with applicability of
the q-Fourier transform in the study of limit processes in nonextensive
statistical mechanics. We prove that the answer is affirmative if and only if q
> 1, excluding two particular cases of q<1, namely, q = 1/2 and q = 2/3, which
are also out of the theory valid for q \ge 1. We also discuss some applications
of the q-Fourier transform to nonlinear partial differential equations such as
the porous medium equation.
