Fourier Transform $\mathcal{F}: L^2(\mathbb{R}) \to L^2(\mathbb{R})$

Question

If we think $\mathcal{F}: L^2(\mathbb{R}) \to L^2(\mathbb{R})$ as the extended Fourier transform. Can we find a function $f$ that is not in $L^1(\mathbb{R})$ but such that $\mathcal{F}f \in L^1(\mathbb{R})$ ?

Answer 1

Sure: $f(x) = \dfrac{\sin x}{x},$ which is in $L^2$ but not in $L^1,$ and whose $L^2$ Fourier transform is, up to a constant multiple, $\chi_{[-1,1]}.$