1301.3518 (A. Plastino et al.)

q-Fourier Transform: reconciling Hilhorst with Umarov-Tsallis-Steinberg    [PDF]

A. Plastino, M. C. Rocca

q-Fourier transforms, developed by Umarov-Tsallis-Steinberg [Milan J. Math. {\bf 76} (2008) 307], mightily contribute to cement the foundations of non-extensive statistical mechanics. Recently, Hilhorst [J. Stat. Mech. (2010) P10023] investigated the feasibility of obtaining an invertible q-Fourier transformation (qFT) by restricting the domain of action of the transform to a suitable subspace of probability distributions and showed that this is impossible. Further, by explicit construction, he encountered families of functions, all having the same qFT (the q-Gaussians themselves being part of such families) and showed in that important case the non-invertibility of the qFT. In this communication we show that, although Hilhorst is right, the qFT does indeed map, in a one-to-one fashion, {\it classes} of functions into other classes, not isolated functional instances. Thus, the foundational qFT debate seems to have been resolved.

View original: http://arxiv.org/abs/1301.3518