# $q$-Generalized representation of the $d$-dimensional Dirac delta and $q$-Fourier transform

## TL;DR

This paper introduces a $q$-generalized representation of the Dirac delta in multiple dimensions and explores its application to the $q$-Fourier transform, demonstrating invertibility and effects on the Gibbs phenomenon.

## Contribution

It presents a novel $q$-generalized Dirac delta representation and analyzes the invertibility of the $q$-Fourier transform in $d$ dimensions.

## Key findings

- $q$-delta representation is valid in $d$ dimensions
- $q$-Fourier transform is invertible for all $d$
- $q$-deformation influences the Gibbs phenomenon

## Abstract

We discuss a generalized representation of the Dirac delta function in $d$ dimensions in terms of $q$-exponential functions. We apply this new representation to the study of the so-called $q$-Fourier transform, proving its invertibility for any value of $d$. We finally illustrate the effect of the $q$-deformation on the Gibbs phenomenon.