Can $f\notin L^1$ if its Fourier transform $\hat f \in L^\infty$ ?
This is a question about the endpoint of Pontryagin duality. We know that if a function is in $L^1$, then its Fourier transform lies in $L^\infty$. This is very easy to show. But what about the converse: Can the $L^1$ norm of $f$ be infinite even if the $L^\infty$ norm of its Fourier transform is finite?
I assume that the answer to my question is YES, but I do not see how to handle this case. Any references?