  The final goal of the present work is to extend the Fourier transform on the
Heisenberg group $\H^d,$ to tempered distributions. As in the Euclidean
setting, the strategy is to first show that the Fourier transform is an
isomorphism on the Schwartz space, then to define the extension by duality. The
difficulty that is here encountered is that the Fourier transform of an
integrable function on $\H^d$is no longer a function on $\H^d$ : according to
the standard definition, it is a family of bounded operators on $L^2(\R^d).$
Following our new approach in\ccite{bcdFHspace}, we here define the Fourier
transform of an integrable functionto be a mapping on the
set~$\wt\H^d=\N^d\times\N^d\times\R\setminus\{0\}$endowed with a suitable
distance $\wh d$.This viewpoint turns out to provide a user friendly
description of the range of the Schwartz space on $\H^d$ by the Fourier
transform, which makes the extension to the whole set of tempered distributions
straightforward. As a first application, we give an explicit formula for the
Fourier transform of smooth functions on $\H^d$ that are independent of the
vertical variable. We also provide other examples.
