On a representation of the inverse Fq transform

TL;DR

This paper derives a representation formula for the inverse q-Fourier transform within a specific function class, advancing the mathematical foundation for applications in nonextensive statistical mechanics.

Contribution

It provides the first explicit representation of the inverse q-Fourier transform for functions in a defined class, facilitating future theoretical and practical applications.

Findings

Representation formula for inverse q-Fourier transform derived

Applicable to functions in the class _q

Lays groundwork for further applications in physics

Abstract

A recent generalization of the Central Limit Theorem consistent with nonextensive statistical mechanics has been recently achieved through a generalized Fourier transform, noted $q$-Fourier transform. A representation formula for the inverse $q$-Fourier transform is here obtained in the class of functions $\mathcal{G}=\bigcup_{1\le q<3}\mathcal{G}_q,$ where $\mathcal{G}_{q}=\{f = a e_{q}^{-\beta x2}, \, a>0, \, \beta>0 \}$. This constitutes a first step towards a general representation of the inverse $q$-Fourier operation, which would enable interesting physical and other applications.