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

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.