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