Say we have a function $f:\mathbb{R}^n \to \mathbb{R}$ such that $$ f(x) = \int_{\mathbb{R}^n} \psi(k) e^{ik \cdot x} dk $$ where $\psi$ is the Fourier dual of $f$. Say we further know that $\psi > 0$ on $\mathbb{R}^n$, and therefore by Bochner's theorem, $f$ is positive definite. Is there a method to deduce the minimum of $f$ using $\psi$?
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$\begingroup$ since a positive definite function satisfies $|f(x)|\leq f(0)$, its minimum is $-\int\psi(k)\,dk$. $\endgroup$– Carlo BeenakkerCommented Apr 24, 2022 at 21:11
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$\begingroup$ @CarloBeenakker Thank you for your reply. I can see why this is a lower bound, but I cannot see why this would imply it to be a minimum. Could you elaborate? $\endgroup$– spacemanCommented Apr 24, 2022 at 21:13
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1$\begingroup$ you're right, it might not reach this lower bound. $\endgroup$– Carlo BeenakkerCommented Apr 24, 2022 at 21:17
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1$\begingroup$ In general the minimum can be any number strictly between $-1$ and $1=\int \psi$. $\endgroup$– Giorgio MetafuneCommented Apr 25, 2022 at 13:32
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