On a space of entire functions and its Fourier transformation

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 a help of a family of weight functions (not radial in general) is considered in the paper. For this space an equivalent description in terms of estimates on all partial derivatives of functions on ${\mathbb{R}}^n$ and Paley-Wiener type theorem are given.