k-Deformed Fourier Transform

TL;DR

This paper introduces a $5$-deformed Fourier transform within $5$-algebra, extending Fourier analysis to $5$-generalized statistical mechanics, and explores its properties and applications such as a $5$-central limit theorem.

Contribution

It formulates a new $5$-Fourier transform based on $5$-algebra, linking it to standard Fourier analysis and applying it to $5$-statistics.

Findings

The $5$-Fourier transform reduces to the standard Fourier transform as $5 o 0$.

The $5$-transform exhibits wavelet-like behavior due to damping factors.

A $5$-central limit theorem for $5$-independent variables is discussed.

Abstract

We present a new formulation of Fourier transform in the picture of the $\kappa$-algebra derived in the framework of the $\kappa$-generalized statistical mechanics. The $\kappa$-Fourier transform is obtained from a $\kappa$-Fourier series recently introduced by us [2013 Entropy {\bf15} 624]. The kernel of this transform, that reduces to the usual exponential phase in the $\kappa\to0$ limit, is composed by a $\kappa$-deformed phase and a damping factor that gives a wavelet-like behavior. We show that the $\kappa$-Fourier transform is isomorph to the standard Fourier transform through a changing of time and frequency variables. Nevertheless, the new formalism is useful to study, according to Fourier analysis, those functions defined in the realm of the $\kappa$-algebra. As a relevant application, we discuss the central limit theorem for the $\kappa$-sum of $n$-iterate statistically independent random variables.