  Using the existence of infinite numbers $k$ in the non-Archimedean ring of
Robinson-Colombeau, we define the hyperfinite Fourier transform (HFT) by
considering integration extended to $[-k,k]^{n}$ instead of
$(-\infty,\infty)^{n}$. In order to realize this idea, the space of generalized
functions we consider is that of generalized smooth functions (GSF), an
extension of classical distribution theory sharing many nonlinear properties
with ordinary smooth functions, like the closure with respect to composition, a
good integration theory, and several classical theorems of calculus. Even if
the final transform depends on $k$, we obtain a new notion that applies to all
GSF, in particular to all Schwartz's distributions and to all Colombeau
generalized functions, without growth restrictions. We prove that this FT
generalizes several classical properties of the ordinary FT, and in this way we
also overcome the difficulties of FT in Colombeau's settings. Differences in
some formulas, such as in the transform of derivatives, reveal to be meaningful
since allow to obtain also non-tempered global unique solutions of differential
equations.
