The Wiener algebra $\mathcal W$ is defined as
$\text{Fourier}(L^1(\mathbb R))$, i.e. the image by the Fourier transform of $L^1(\mathbb R)$. Riemann-Lebesgue's lemma ensures that 
$$
\mathcal W\subset C^0_{(0)}(\mathbb R)=\{\phi\text{ continuous on }\mathbb R, \lim_{\vert \xi\vert\rightarrow+\infty}\phi(\xi)=0\}.
$$
(1) I believe that the injection $\mathcal W\subset C^0_{(0)}(\mathbb R)$ is not onto (is it due to Hardy? Gaier? Both at different times?).

(2)
Is there an "explicit" function $\phi\in C^0_{(0)}(\mathbb R)$ whose inverse Fourier transform (say in the distribution sense) does not belong to  $L^1(\mathbb R)$? 

(3) Is there a functional analysis reason for which the Banach spaces $L^1(\mathbb R)$ and  $C^0_{(0)}(\mathbb R)$ cannot be isomorphic?