Function $f$ with $\hat{f}=0$

Question

Is there a function $0\neq f\in L^1(\mathbb R^d) $ with $\hat{f}=0$?

Answer 1

No, if that were the case, $f,\hat{f}\in L^1(\mathbb{R}^d)$. Thus, the Fourier Inversion Theorem holds (see Theorem 8.26 Folland "Real Analysis"), so $f$ is almost everywhere equal to zero.