**N.B.** this answer was in response to an earlier version of the question, which only had the first two paragraphs -- hence it doesn't address what appears to have been the original poster's *actual* question. For that, see the answers of Leonid or Harald.

I'm not sure if this answers your question, but it might be worth noting that a measurable function $f$ on the real line is in $L^2({\mathbb R})$ if and only if its Fourier transform $\widehat{f}$ is (Plancherel theorem), while it is in $C^\infty({\mathbb R})$ if and only if we have

$$ \int_{-\infty}^\infty | \widehat{f}(x) |^2 (1+ |x|^2)^{k} < \infty \quad{\rm for }\ k=1,2,\dots $$

(this is a form of Sobolev embedding, albeit in a very special case). In particular, if I've correctly understood the notation from the wikipedia page for Sobolev spaces, the space you're after seems to be the intersection $\bigcap_{k=0}^\infty H^k({\mathbb R})$. I don't know if this goes by a particular name.