q-Generalization of the inverse Fourier transform

TL;DR

This paper introduces a method to uniquely recover distributions from their q-Fourier transforms, extending classical Fourier inversion to the q-generalized context, aiding the analysis of complex systems with correlated variables.

Contribution

It presents a novel inverse q-Fourier transform method using a q-generalized Dirac delta, extending classical Fourier inversion for complex, correlated systems.

Findings

Method unambiguously determines distributions from q-Fourier transforms.

Extends classical Fourier inverse to q-generalized setting.

Potential applications in complex systems analysis.

Abstract

A wide class of physical distributions appears to follow the q-Gaussian form, which plays the role of attractor according to a Central Limit Theorem generalized in the presence of specific correlations between the relevant random variables. In the realm of this theorem, a q-generalized Fourier transform plays an important role. We introduce here a method which univocally determines a distribution from the knowledge of its q-Fourier transform and some supplementary information. This procedure involves a recently q-generalized Dirac delta and the class of functions on which it acts. The present method conveniently extends the inverse of the standard Fourier transform, and is therefore expected to be very useful in the study of many complex systems.