Is anyone aware of an explicit example of a (Schwartz, real-analytic, extending to an entire function with suitable decay properties along imaginary directions as for the Paley–Wiener-Schwartz theorem) function $\mathbb{R} \to \mathbb{C}$ whose Fourier transform is in $C^\infty_0(\mathbb{R}; \mathbb{C})$?
I'm hoping to find one such example written in a form simple enough to be suitable for computation (not as an inverse Fourier transform of a $C^\infty_0$ function!).
I found somewhere the claim that $$ f(x) = \frac{\sin(\pi x)}{\sin(\pi \sqrt x) \sinh(\pi \sqrt x))} $$ works, but sadly I'm unable to verify this. Thanks for your help!