Image of Fourier Transform

In analysis today, we were talking about the fourier transform and how as a map from L_2 to L_2 it is onto, but as a map from L_1 to C_0 it isn’t onto and the range isn’t even closed. Then Dr. Jury said that the range is really hard to describe.

So after class I asked him if “hard to describe” meant “analytic but non-Borel” since the range is analytic by definition. He didn’t know off-hand. I did a quick look online and didn’t find other than people saying that the range is hard to characterize.

So, is there any result saying it is \Sigma^1_1 complete or at least non-Borel? I would be interested to know, and to see the proof.  

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