I am trying to understand, because the Fourier transform of the function $f(x) = e^{ -\sqrt{ \lvert x \rvert } }$ is smooth.
My question: Under which conditions is the Fourier transform of an $L^1$ or $L^2$ function smooth?
I think $f$ might be in $L^1$ or $L^2$, but I wasn't able to find a good way to determine that yet. I know that the Fourier transform of a function with compact support is smooth and that if a function is $k$-times differentiable, then the follwing holds for the functions fourrier transform: $$ \hat{f}(x) \leq \frac{C}{(1+\lvert x \rvert )^{k}} $$ From the comments and discussion with friends using it appears you have to used the Dominated convergence theorem to exchange derivative and integral, but I struggle to find the dominating function.