TITLE: Range of the Fourier transform on $L^1$
QUESTION [27 upvotes]: It is well known that the Fourier transform $\mathcal{F}$ maps 
$L^1(\mathbb{R}^d)$ into, but not onto, $\overline{C_0^0}(\mathbb{R}^d)$, where the closure is taken in the $L^\infty$ norm. This is a consequence of the open mapping theorem, for instance. 
My question is: what's an explicit example of a function in $\overline{C_0^0}(\mathbb{R}^d)$ which is not in the image of $L^1(\mathbb{R}^d)$ under the Fourier transform? 
I would also like to know whether there is a useful characterization of $\mathcal{F}(L^1(\mathbb{R}^d))$. 
Remark: it is easy to see that the Banach space $\overline{C_0^0}(\mathbb{R}^d)$ consists of all continuous functions $f$ on $\mathbb{R}^d$ such that $f(\xi)\rightarrow 0$ as $|\xi|\rightarrow\infty$.

REPLY [4 votes]: See Page 68 (Theorem 3.2.2) in my notes on Fourier Analysis:
https://home.iitm.ac.in/mtnair/FS-2018.pdf