Is the F.T of $\operatorname{Span}(\mathscr S(\mathbb R^2)\otimes\mathscr D(D))$, $D\subset\mathbb R^2$ dense in $L^2(\mathbb R^4)$?

Question

Let $K$ be the real vector space generated by elements $f$ in $\mathscr S(\mathbb R^2,\mathbb R)\otimes\mathscr D(D, \mathbb R)$, where $D$ is any bounded subset of $\mathbb R^2$. Let $\hat K$ be the vector space generated by the Fourier transform of each $f\in K$, i.e. $\hat K = \{\hat f\ |\ f\in K\}$. Is $\hat K+i\hat K$ dense in $L^2(\mathbb R^4)$?

Answer 1

I'm sorry if this is not the appropriate place to ask such questions. I'll be more careful next time. By the way my question is strongly related to an actual problem in QFT (actually QFT on QST). I'm not sure about M. Bischoff conclusion on orthogonality pointed out in one of the comments to the question.

There is a result from Araki that state that the real Hilbert subspace of the one-particle Hilbert space $K(O)=\{\hat f\big\vert_{\Omega_m^+}\ |\ f\in\mathscr D(O,\mathbb R)\}$, where $\Omega_m^+$ is the hyperboloid of mass $m$ in the future light-cone and $O\subset\mathbb R^4$ is a non-empty simply connected bounded open subset of $\mathbb R^4$, is standard, i.e. $\overline{K(O)+iK(O)}$ is dense and $K\cap iK=\{0\}$. The vector space $\hat K$ I defined in the question ought to contain such a vector space $K(O)$ associated with a region $O\subset\mathbb R^2\times D$, where $D$ satisfies some suitable regularity condition (e.g. regular boudary, simply connectedness,...) and therefore $K(O)\subset \hat K$, that would imply $\hat K$ standard, according to the result from Araki.