limits respectively of $\mathcal S_{s,h}({\mathbf R^{d}})$ with respect to $h$. This implies that $$\label{GSspacecond1}
\mathcal S_s({\mathbf R^{d}}) = \bigcup _{h>0}\mathcal S_{s,h}({\mathbf R^{d}})
\quad \text{and}\quad \Sigma _{s}({\mathbf R^{d}}) =\bigcap _{h>0}\mathcal
S_{s,h}({\mathbf R^{d}}),$$ and that the topology for $\mathcal S_s({\mathbf R^{d}})$ is the strongest possible one such that each inclusion map from $\mathcal S_{s,h}({\mathbf R^{d}})$ to $\mathcal S_s({\mathbf R^{d}})$ is continuous. The space $\Sigma _s({\mathbf R^{d}})$ is a Fr[é]{}chet space with semi norms ${\Vert {\, \cdot \, }\Vert _{\mathcal S_{s,h}}}$, $h>0$. Moreover, $\mathcal S _s({\mathbf R^{d}})\neq \{ 0\}$, if and only if $s\ge 1/2$, and $\Sigma _s({\mathbf R^{d}})\neq \{ 0\}$, if and only if $s>1/2$.

For every ${\varepsilon}>0$ and $s>0$, we have $$\Sigma _s ({\mathbf R^{d}})\subseteq \mathcal S_s({\mathbf R^{d}})\subseteq
\Sigma _{s+{\varepsilon}}({\mathbf R^{d}}).$$

The *Gelfand-Shilov distribution spaces* $\mathcal S_s'({\mathbf R^{d}})$ and $\Sigma _s'({\mathbf R^{d}})$ are the projective and inductive limit respectively of $\mathcal S_{s,h}'({\mathbf R^{d}})$. Hence $$\tag*{(\ref{GSspacecond1})$'$}
\mathcal S_s'({\mathbf R^{d}}) = \bigcap _{h>0}\mathcal
S_{s,h}'({\mathbf R^{d}})\quad \text{and}\quad \Sigma _s'({\mathbf R^{d}})
=\bigcup _{h>0} \mathcal S_{s,h}'({\mathbf R^{d}}).$$ By [@Pil], $\mathcal S_s'$ and $\Sigma _s'$ are the duals of $\mathcal S_s$ and $\Sigma _s$, respectively.

The Gelfand-Shilov spaces are invariant or posses convenient mapping properties under several basic transformations. For example they are invariant under translations, dilations, 