Fourier transforms of rapidly decreasing functions

New spaces of rapidly decreasing infinitely differentiable functions on ${\mathbb R}^n$ will be defined in the talk. They are introduced with a help of a family of separately radial convex functions on ${\mathbb R}^n$ satisfying some technical conditions and very similar in construction to Gelfand–Shilov spaces $S_{\alpha}$. For all of them Paley–Wiener type theorems are obtained. Using some recent facts of convex analysis it will be shown that the Gelfand-Shilov space $W_M$ is a particular case of one of these spaces.