does anybody know an example of a real function $f\in C_c^\infty(\mathbb R)\setminus\{0\}$ (non-zero, smooth with compact support) such that for its fourier transform $\hat f(\xi)=\int_\mathbb R f(x)e^{-2\pi i x\xi}~dx$ a closed/explicit form (without integral) is known?
Or the other way around: Does somebody know an explicitly represented schwartz function $g\neq 0$ on $\mathbb R$ such that $\hat g\in C_c^\infty(\mathbb R)$?
I've tried some constructions with $e^{-1/x^2}$, but the occurring integrals aren't solvable (at least for me and maple ;)).
Thank you.