# Zero set of the Fourier transform of measures

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.

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

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'.