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Let $\mu$ be a probability measure on $\mathbb R$ and the Fourier transform of $\mu$ is defined by $$ \widehat{\mu}(\xi) = \int_{\mathbb{R}} e^{-2\pi i \xi x} ~ d \mu(x). $$ Let $\mathcal{Z}(\widehat{\mu})=\{ \xi \in \mathbb R: \widehat{\mu}(\xi)=0 \}$.

Question: how about the set $\mathcal{Z}(\widehat{\mu})$? Can we show that the set $\mathcal{Z}(\widehat{\mu})$ is at most countable?


Try: when $\mu$ has compact support, we can extend $\widehat{\mu}$ to the whole complex plane by $$ \widehat{\mu}(z) = \int_{\mathbb{R}} e^{-2\pi i z x} ~ d \mu(x). $$ Next, using compact support, we can show $\widehat{\mu}(z)$ is an entire function. By the theorem in complex analysis, we can obtain that the set $\mathcal{Z}(\widehat{\mu})$ has no limit point.

I have no idea of the case when the support of $\mu$ is not compact.

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  • $\begingroup$ The characteristic function of probability measure can vanish throughout some inerval. Such examples are available in Feller's book. $\endgroup$ Mar 9, 2020 at 10:20
  • $\begingroup$ @KaviRamaMurthy can you give the name of Feller’s book? $\endgroup$
    – wzstrong
    Mar 9, 2020 at 10:22
  • $\begingroup$ I have posted an answer. $\endgroup$ Mar 9, 2020 at 10:26

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The function $\phi (t)=1-|t|$ for $|t| \leq 1$ and $0$ for $|t| >1$ is the Fourier transform of a probablity measure. This is proved in Feller's book: An Introduction to Probability Theory and its Applications. See the section on 'Special densities. Mixtures' in the chapter on 'Characterisitic Functions'.

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