q-Fourier Transform and its inversion-problem
  Tsallis' q-Fourier transform is not generally one-to-one. It is shown here that, if we eliminate the requirement that $q$ be fixed, and let it instead "float", a simple extension of the $F_q-$definition, this procedure restores the one-to-one character. 
A. Plastino and M.C.Rocca
math-ph cond-mat.stat-mech math.MP