Fourier Transforms of Rapidly Decreasing Functions on ℝ^n

With help of a certain family ℋ of separately radial convex functions in ℝ^n two new spaces of rapidly decreasing infinitely differentiable functions in ℝ^n are introduced in the article. One of them, namely, the space G(ℋ) , is a subspace of Gelfand–Shilov type space S_α,A , where α=(1,…,1)∈ℝ^n,A=(0,…,0)∈ℝ^n . Functions of the second space ℰ(ℋ) admit an extension to entire functions in ℂ^n . A description of the space of such extensions is obtained. Under some mild additional restrictions on ℋ an isomorphism between the spaces G(ℋ) and ℰ(ℋ) is established via the Fourier transform.