The Fourier Transform $\mathcal{F}:L^1(\mathbb{R})\to C_0(\mathbb{R})$ is an injective, bounded linear map that isn't onto.  It is known (if I remember correctly) that the range isn't closed, but is dense in $C_0$.  Everything I have read/heard says that the range is "difficult to describe".  

But, since $\mathcal{F}$ is injective and continuous, the image of $L^1$ must be Borel inside of $C_0$.  Is anything else known about its descriptive complexity?  If not, might this be an example of a natural set of high Borel rank?

This [question][1] may be relevant. Thanks for any insight/references.


  [1]: http://mathoverflow.net/questions/8085/range-of-the-fourier-transform-on-l1