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

We introduce 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 and we establish the analytical procedure through which it can be inverted for any value of d. We finally illustrate the effect of the q-deformation on the Gibbs phenomenon of Fourier series expansions.