Is there a function $0\neq f\in L^1(\mathbb R^d) $ with $\hat{f}=0$?
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2No. $f \in L^{1}, \hat f=0$ implies $f=0$ a.e. – geetha290krm Aug 19 '22 at 10:03
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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.
Álvaro Romaniega
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