if $f\in L^1(R^n)$ and its Fourier transform $\hat{f}=\int f(x) e^{-ixt} \ dx \in L^2(R^n)$, then can we show $f \in L^2(R^n)$?
If yes, can you tell me why? If not, can you give me some counterexamples? I also see some obscure answers elsewhere with some high-order mathematcical concepts such as fourier transform in schwardz space, I can't understand it. If yes, can you prove it with the knowledge within the scope of Rudin's 'Real and Complex Analysis'?