Fourier Transforms of Rapidly Decreasing Functions on

With help of a certain family of separately radial convex functions in two new spaces of rapidly decreasing infinitely differentiable functions in are introduced in the article. One of them, namely, the space , is a subspace of Gelfand–Shilov type space , where . Functions of the second space admit an extension to entire functions in . A description of the space of such extensions is obtained. Under some mild additional restrictions on an isomorphism between the spaces and is established via the Fourier transform.