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Is it possible to have an even probability measure $\mu$ (that is $\mu(A)=\mu(-A)$ for any set $A\subset \mathbb{R}^d$) supported on the unit sphere $S^{d-1}$ such that its Fourier Transform $$ \widehat{\mu}(\xi) = \int_{S^{d-1}} e^{-2\pi i x\cdot \xi} d\mu(x) $$ is non-negative, that is $\widehat{\mu}(\xi)\geq 0$. If not, then how small $\left|\inf_{\xi \in \mathbb{R}^d} \{\widehat{\mu}(\xi)\}\right|$ can be?

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    $\begingroup$ Average over rotations. $\endgroup$
    – fedja
    Dec 21, 2017 at 1:01
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    $\begingroup$ Hint: If $\mu$ vanishes at 0, what does that imply about $\hat\mu$? $\endgroup$ Dec 21, 2017 at 1:32
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    $\begingroup$ If we average on rotations, we get a measure constant on the sphere, this give us a Bessel function which oscillates a lot and is negative at some points. Your idea actually suggests the question if a uniform measure is indeed the best possible, having the smallest $\inf$. Which I already considered. Also, a priori $\mu$ is supported in the sphere, so "at zero" it does vanishes. $\endgroup$ Dec 22, 2017 at 4:56

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Not to leave this question "unanswered", while the answer (to the qustion in the title) is obvious: because $\mu$ vanishes on a neighborhood of $0$, $\int\hat\mu=0$, so that $\hat\mu$ (real, for even $\mu$) has negative and positive values. Since it is also continuous, it vanishes somewhere.

What seems not obvious is: how small can $-\inf \hat\mu(\xi)$ be?

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    $\begingroup$ Perhaps it is worth emphasizing that the Fourier transform of a compactly supported (regular Borel) measure is indeed a continuous function, so that it makes sense to talk about pointwise values. Thus, approximating $\widehat\mu$ by truncations does give the correct limiting value for $\int \widehat \mu$. $\endgroup$ Dec 23, 2017 at 21:12
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    $\begingroup$ As hinted in the comment by fedja on the original question, an averaging argument shows that $-\inf \hat\mu(\xi)$ is minimized when $\mu$ is the uniform distribution on the sphere, so it comes down to finding the minimum value taken by the relevant Bessel function. $\endgroup$ Dec 24, 2017 at 4:45
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    $\begingroup$ This answers in part the question, I would say. But, for instance, if $\mu$ is a finite sum of deltas on the sphere then $\hat{\mu}$ is a sum of cosines, which is not integrable. This can be fixed by an approximation of the identity argument, showing that $\int \hat{\mu} \hat{\phi}$ is as close to zero as one wishes by choosing $\phi$ close to a delta at zero with $\hat{\phi}\geq 0$ and close to $1$. But, I still do not know how to show that the argument over rotations would imply that the the constant measure (producing a bessel) is the one with the smallest $\inf \hat{\mu(\xi)}$. $\endgroup$ Dec 24, 2017 at 20:35

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