If $f$ is smooth and the inverse Fourier transform is zero

Question

(1)If $f$ is smooth and the inverse Fourier transform is zero that $$(f)^{\lor}(\xi)=0$$ Can we say $\hat{f}$ is real or other properties about $f$?

(2)If $\int f=0$, how about the $\hat{f}$?

Answer 1

Yes, for $f$ a tempered distribution, so that $\widehat{f}$ makes sense as a tempered distribution, and is the $0$ distribution, then $f$ must be zero. The smoothness plays no role.

Saying $\int f=0$ is ambiguous without further info about $f$, even for smooth $f$, since $f$ certainly need not be in $L^1$, so the literal integral expression the Fourier transform is ambiguous, and it is not at all the case the every such $f$ (smooth or not) gives a tempered distribution... so it's not clear what the Fourier transform of such $f$ would be.

(And then, certainly if $f=0$ as a distribution, and if $f$ is smooth, it has pointwise values, which are necessarily all $0$, yes.)