Let $\mu$ be a probability measure on $\mathbb{R}$. Is the following implication true? $$ \widehat{\mu}(y) \rightarrow 0 \text{ as } |y| \rightarrow \infty \quad \Rightarrow \quad \mu(\{x\})=0 \quad \forall x\in \mathbb{R} $$
2026-03-31 21:10:40.1774991440
If the Fourier transform of a probability measure goes to zero at infinity, can the measure have a point mass?
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According to these notes (page 3), this was proved by Neder in 1920. One can also use the criterion given by Wiener in 1924, reminded in the same notes.
This is about measures on the circle, but I believe there must not be a big difference with the case of the real line.