Selfanswer!
The structure of $\mathscr{H}$ is closely related to the structure of the RKHS $\mathscr{G}$ generated by the Gaussian kernel and $\mathscr{G}$ has been thoroughly analysed.
Relation to the Gaussian kernel
It is straightforward to verify that the kernels $K(x,y)=\exp\left\{x y\right\}$ of $\mathscr{H}$ and $G(x,y)=\exp\left\{-\frac{1}{2}(x - y)^2\right\}$ of $\mathscr{G}$ are related by:
$$ K(x,y) =f(x) G(x,y) f(y) \text{ with } f(x) = \exp\left\{\frac{1}{2}x^2\right\}.$$
Relation between $\mathscr{H}$ and $\mathscr{G}$
From the relation above between the kernels follow the following facts about the spaces and the scalar product (see e.g. Paulsen Proposition 5.20):
- $\mathscr{H}=\left\{fg\mid g \in \mathscr{G}\right\}$
- $<fg_1, fg_2>_\mathscr{H} \,=\, <g_1,g_2>_\mathscr{G}.$
Facts about $\mathscr{G}$ and conclusions
In Steinwart-Christmann Theorem 4.42 $\mathscr{G}$ is characterised by:
- The functions $\tilde{e}_n=\frac{1}{\sqrt{n!}}x^n\exp\left\{-\frac{1}{2}x^2\right\}$ are an ONB of $\mathscr{G}.$
- $\mathscr{G}=\left\{g=\sum{\alpha_n \tilde{e_n}\mid \sum \alpha_n^2 < \infty}\right\}$
and one can conclude about $\mathscr{H}:$
- The functions $e_n=\frac{1}{\sqrt{n!}}x^n$ are an ONB of $\mathscr{H}.$
- $\mathscr{H}=\left\{h=\sum{\frac{\alpha_n }{\sqrt{n!}}x^n\mid \sum \alpha_n^2 < \infty}\right\}$.
The series can be compared to the Taylor expansion of $h\in\mathscr{H}$ to conclude that $\mathscr{H}$ consists of real analytic functions such that
$$ \sum{\frac{(h^{(n)}(0))^2}{n!} }<\infty.$$
