{\mathscr P}_E({\mathbf R^{d}})$, there is a constant $r>0$ such that $$\label{WeightExpEst}
e^{-r|x|}\lesssim \omega (x)\lesssim e^{r|x|},\quad x\in {\mathbf R^{d}}$$ (cf. [@Gc2.5]).

We say that $v$ is *submultiplicative* if $v$ is even and holds with $\omega =v$. In the sequel, $v$ and $v_j$ for $j\ge 0$, always stand for submultiplicative weights if nothing else is stated.

Gelfand-Shilov spaces
---------------------

Next we recall the definition of Gelfand-Shilov spaces.

Let $h,s\in \mathbf R_+$ be fixed. Then $\mathcal S_{s,h}({\mathbf R^{d}})$ is the set of all $f\in C^\infty ({\mathbf R^{d}})$ such that $${\Vert f\Vert _{\mathcal S_{s,h}}}\equiv \sup \frac {|x^\beta \partial ^\alpha
f(x)|}{h^{|\alpha + \beta |}(\alpha !\, \beta !)^s}$$ is finite. Here the supremum is taken over all $\alpha ,\beta \in
\mathbf N^d$ and $x\in {\mathbf R^{d}}$.

Obviously $\mathcal S_{s,h}({\mathbf R^{d}})$ is a Banach space which increases as $h$ and $s$ increase, and is contained in ${\mathscr S}({\mathbf R^{d}})$, the set of Schwartz functions on ${\mathbf R^{d}}$. Furthermore, if $s>1/2$, or $s=1/2$ and $h$ is sufficiently large, then is dense in $\mathscr S$. Hence, the dual $(\mathcal S_{s,h})'({\mathbf R^{d}})$ of $\mathcal S_{s,h}({\mathbf R^{d}})$ is a Banach space which contains $\mathscr S'({\mathbf R^{d}})$.

The *Gelfand-Shilov spaces* $\mathcal S_{s}({\mathbf R^{d}})$ and $\Sigma _s({\mathbf R^{d}})$ are the inductive and projective 