The $\tau_q$-Fourier transform: covariance and uniqueness

TL;DR

The paper introduces the $ au_q$-Fourier transform, inspired by Tsallis entropy, highlighting its covariance properties and demonstrating its automatic invertibility and essential uniqueness under certain algebraic assumptions.

Contribution

It proposes a new Fourier transform based on Tsallis entropy principles, analyzing its covariance, invertibility, and uniqueness in the context of algebraic fields.

Findings

The $ au_q$-Fourier transform is automatically invertible in proper contexts.

It is essentially unique under the exchange of point-wise product and convolution.

The transform is motivated by covariance properties related to algebraic fields.

Abstract

We propose an alternative definition for a Tsallis entropy composition-inspired Fourier transform, which we call "$\tau_q$-Fourier transform". We comment about the underlying "covariance" on the set of algebraic fields that motivates its introduction. We see that the definition of the $\tau_q$-Fourier transform is automatically invertible in the proper context. Based on recent results in Fourier analysis, it turns that the $\tau_q$-Fourier transform is essentially unique under the assumption of the exchange of the point-wise product of functions with their convolution.