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 L2(mathbbR) if and only if its Fourier transform widehatf is (Plancherel theorem), while it is in Cinfty(mathbbR) if and only if we have
inti−inftynfty|widehatf(x)|2(1+|x|2)k<inftyquadrmfork=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 bigcapik=0nftyHk(mathbbR). I don't know if this goes by a particular name.
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