Reflections on the -Fourier transform and the -Gaussian function

The standard q -Fourier Transform (qFT) of a constant diverges, which begs for a better treatment. In addition, Hilhorst has conclusively proved that the ordinary qFT is not of a one-to-one character for an infinite set of functions [H.J. Hilhorst, J. Stat. Mech. (2010) P10023]. Generalizing the ordinary qFT analyzed in [S. Umarov, C. Tsallis, S. Steinberg, Milan J. Math. 76 (2008) 307], we appeal here to a complex q -Fourier transform, and show that the problems above mentioned are overcome.