Image of L^1 under the Fourier Transform 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 may be relevant. Thanks for any insight/references.
 A: It seems that Mike's argument shows that in fact the range $R$ of the Fourier transform is $F_{\sigma\delta}$ in $C_0(\mathbb R)$. Define the $T_ng$ as above, 
$$T_ng(x)=\int_{\mathbb R} e^{-a_n\pi\vert t\vert}g(t) e^{2i\pi tx}dt\;.$$ Then a function $g\in C_0(\mathbb R)$ is in $R$ iff two things hold:
(i) all $T_ng$ are in $L^1$;
(ii) the sequence $(T_ng)$ is Cauchy in $L^1$.
Condition (i) can be written as follows:
$$ \forall n\;\exists N\in\mathbb N\; \left ( \forall d\in\mathbb R_+ \;:\;\int_{-d}^d \vert T_ng(x)\vert dx\leq N\right)$$
By dominated convergence, the condition under brackets is closed with respect to $g$; so (i) defines an $F_{\sigma\delta}$ subset of $C_0(\mathbb R)$.
Condition (ii) reads
$$\forall k\in\mathbb N\;\exists N \;\left(\forall p,q\geq N\; \forall d \;:\; \int_{-d}^d \vert T_pg(x)-T_qg(x)\vert dx\leq \frac 1k\right)$$
By dominated convergence again, the condition under brackets is closed wrt $g$, so (ii) defines an $F_{\sigma\delta}$ subset of $C_0(\mathbb R)$.
Altogether, $R$ is the intersection of two $F_{\sigma\delta}$ sets, hence an $F_{\sigma\delta}$ subset of $C_0(\mathbb R)$. I would be extremely surprised if it were better than that; i.e.   I  "conjecture" that it is not $G_{\delta\sigma}$.
A: Here is an attempt, somewhat rough around the edges but I think it works:
Claim: The range of the Fourier transform $\mathcal{F}:L^1(\mathbb
R)\to C_0(\mathbb R)$ is a Borel set in $C_0(\mathbb R)$ of the form
\begin{equation}
\bigcap_{k=1}^\infty \bigcup_{N=1}^\infty \bigcap_{m,n\geq N} E_{m,n,k}
\end{equation}
where each $E_{m,n,k}$ is an $F_\sigma$.
Proof: Consider the cutoff functions $\{e^{-a\pi|t|}\}$ and fix a
sequence $a_n\to 0$. It is a fact that a function $g\in C_0(\mathbb
R)$ is the Fourier transform of some $f\in L^1$ if and only if the
sequence
\begin{equation}
  T_n(g)(x) := \int_{\mathbb R} e^{-a_n\pi |t|} g(t)e^{2\pi itx} dt
\end{equation}
is Cauchy in $L^1$, in which case if we put $f=\lim T_ng$ then $g
=\widehat{f}$.  Let $R$ denote the range of the Fourier transform in
$C_0$.  If we define
\begin{equation}
  E_{n,m,k} = \lbrace g\in C_0: T_ng, T_mg\in L^1 ,{||T_n(g)-T_m(g)||}_1 \leq\frac{1}{k} \rbrace
\end{equation}
then $R$ has the claimed form.
To prove that $E_{m,n,k}$ is an $F_\sigma$,  put $T_n(g)-T_m(g):=T_{mn}(g)$, explicitly
\begin{equation}
 T_{mn}g(x):= T_ng(x)-T_mg(x) =\int_{\mathbb R} h_{mn}(x,t)g(t) dt
\end{equation}
where we define
\begin{equation}
  h_{mn}(x,t) = (e^{-a_m\pi|t|} - e^{-a_n\pi|t|} )e^{2\pi itx}
\end{equation}
Note that by monotone convergence, $g\in E_{m,n,k}$ if and only if two conditions are satisfied: first, for fixed $n$, we need $T_ng\in L^1$. By monotone convergence this is equivalent to: There exists an integer $N$ such that for all integers $d\geq 1$,
\begin{equation}
\left\| {\bf 1}_{[-d,d]}(x)T_ng(x)\right\|_1 \leq N.
\end{equation}
By dominated convergence the set of all such $g$ obeying this for fixed $N,n,d$ is closed, so the set that obeys this for some $N$ and all $d$ (with $n$ held fixed) is an $F_\sigma$.
Thus for fixed $m,n$ the set of $g$ for which $T_ng, T_mg\in L^1$ is an $F_\sigma$.
Additionally, for all integers $d\geq 1$, we need the condition
\begin{equation}
  \left\| {\bf 1}_{[-d,d]}(x) T_{mn} g(x)\right\|_1 \leq \frac{1}{k}.
\end{equation}
But similarly, the set of $g$ obeying this for fixed $m,n,k,d$ is closed in $C_0(\mathbb R)$, and we conclude that each $E_{m,n,k}$ is an $F_\sigma$.
