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 analytic reason for why the Banach spaces $L^1(\mathbb R)$ and $C^0_{(0)}(\mathbb R)$ cannot be isomorphic?