k-Deformed Fourier Transform

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

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