A new space for fourier transform

In this paper we extend the classical theory of the Fourier transofrm on the space of tempered distributions. We construct a testing function space which is contained and dense in the space of rapidly decreasing functions. We introduce and study thepropertioes of the Fourier transform on this testing function space as a prelude to the introduction of the Fourier transform on its dual. We also obtain an inversinon theorem for this Fourier transform. These investigations enable us to obtain a representation theorem for the elements of the dual of this space.