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Suppose $f:\mathbb{R} \rightarrow \mathbb{R}$ is square integrable, symmetric, has infinite support ($\text{supp}(f)= \mathbb{R}\backslash U$, where $U$ is a set of points), and decays at infinity.

Is it true (or false) that

if the Fourier transform of $f$ also has infinite support, then the variance of the distribution $\hat f =|f|^2/\int_{-\infty}^{\infty} |f|^2$ is well-defined ($\int_{-\infty}^{\infty} x^2 \hat f(x) \, dx < +\infty$) ?

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