On the nature of the Tsallis-Fourier Transform

TL;DR

This paper explores the mathematical properties of the q-Fourier transform within Tsallis' q-statistics, revealing its mapping of function classes and addressing its invertibility issues using tempered ultradistributions.

Contribution

It demonstrates that the q-Fourier transform maps equivalence classes of functions into other classes in a one-to-one manner, providing a new perspective on Tsallis' q-statistics.

Findings

qFT maps equivalence classes of functions

Addresses non-invertibility of qFT

Identifies an open problem remaining in the theory

Abstract

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.