# 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.

1. After thinking about it a little more, I remembered that since the Fourier transform is injective, the image of $L_1$ must at least be Borel. But since it isn’t “easily characterized” it must not be at a low level, or at least not obviously.