On Gelfand–Shilov spaces

In this work we follow the scheme of constructing of Gelfand–Shilov spaces $S_{\alpha}$ and $S^{\beta}$ by means of some family of separately radial weight functions in ${\mathbb R}^n$ and define two spaces of rapidly decreasing infinitely differentiable functions in ${\mathbb R}^n$. One of them, namely, the space ${\mathcal S}_{\mathcal M}$ is an inductive limit of countable-normed spaces
\begin{equation*} {\mathcal S}_{\mathcal M_{\nu}} = \bigg\{f \in C^{\infty}({\mathbb{R}}^n): \Vert f \Vert_{m, \nu} = \sup_{x \in {\mathbb{R}}^n, \beta \in {\mathbb{Z}}_+^n, \atop \alpha \in {\mathbb{Z}}_+^n: \vert \alpha \vert \le m} \frac {\vert x^{\beta}(D^{\alpha}f)(x) \vert}{\mathcal M_{\nu}(\beta)} < \infty, m \in {\mathbb{Z}}_+ \bigg\}. \end{equation*}
Similarly, starting with the normed spaces
\begin{equation*} {\mathcal S}_m^{\mathcal M_{\nu}} =\bigg\{f \in C^{\infty}({\mathbb{R}}^n): \rho_{m, \nu}(f) = \sup_{x \in {\mathbb{R}}^n, \alpha \in {\mathbb{Z}}_+^n} \frac {(1+ \Vert x \Vert)^m \vert (D^{\alpha}f)(x) \vert}{\mathcal M_{\nu}(\alpha)} < \infty \bigg\} \end{equation*}
we introduce the space ${\mathcal S}^{\mathcal M}$. We show that under certain natural conditions on weight functions the Fourier transform establishes an isomorphism between spaces ${\mathcal S}_{\mathcal M}$ and ${\mathcal S}^{\mathcal M}$.