Reflections on the q-Fourier transform and the q-Gaussian function

TL;DR

This paper explores a complex q-Fourier transform as a generalization of the real version, using tempered ultra-distributions to effectively derive the q-Gaussian distribution, addressing previous difficulties.

Contribution

It introduces a complex q-Fourier transform framework and demonstrates its effectiveness in deriving the q-Gaussian distribution overcoming prior challenges.

Findings

Successful derivation of q-Gaussian distribution using complex q-Fourier transform

Overcomes issues present in the real q-Fourier transform approach

Utilizes tempered ultra-distributions for mathematical robustness

Abstract

We appeal to a complex q-Fourier transform as a generalization of the (real) one analyzed in [Milan J. Math. {\bf 76} (2008) 307]. By recourse to tempered ultra-distributions we are able to show that the q-Gaussian distribution can be nicely obtained, overcoming all troubles that afflict its real counterpart.