Take the tour ×  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
$L^2(\mathbb R^4)$?  share|improve this question Could you
please clarify the notation? One of the spaces is Schwartz
space, I presume; what's the other one? –  Captain Oates Jun 27
'11 at 9:18 I think the second factor is test functions with
support in D. I am not sure if D is supposed to be fixed or
arbitrary. Either way, this question has the flavor of homework.
–  Michael Renardy Jun 27 '11 at 10:01 I guess $\mathscr D$ are
smooth functions with compact support. If $L^2(\RR^4)$ is the
complex space i guess this is not true, because the
Furiertransform of some real function has some symmetry, and
otherwise the Fourier transform does not make sense to me. – 
Marc