q-Generalization of the inverse Fourier transform

A wide class of physical distributions appears to follow the q-Gaussian form, which plays the role of attractor according to a q-generalized Central Limit Theorem, where a q-generalized Fourier transform plays an important role. We introduce here a method which determines a distribution from the knowledge of its q-Fourier transform and some supplementary information. This procedure involves a recently q-generalized representation of the Dirac delta and the class of functions on which it acts. The present method conveniently extends the inverse of the standard Fourier transform, and is therefore expected to be very useful in the study of many complex systems. ► We present a method to invert the q-Fourier transform of a distribution. ► We illustrate when Dirac delta can be represented using q-exponentials. ► We describe a family of functions for which this new representation works.