On a space of entire functions and its Fourier transform

A space of entire functions of several complex variables rapidly decreasing on ${\mathbb{R}}^n$ and such that their growth along $i{\mathbb{R}}^n$ is majorized with the help of a family of weight functions is considered in this paper. For such space an equivalent description in terms of estimates on all of its partial derivatives as functions on ${\mathbb{R}}^n$ and a Paley-Wiener type theorem are obtained.