= Uncentred Fourier uncertainty principle
{title2=$\|xf\|_2\|f'\|_2\geq\|f\|_2^2/2$}
For nonzero $f$ with $f,xf,f'\in L^2(\mathbb R)$, integration by parts gives $\|f\|_2^2=-2\operatorname{Re}\langle f',xf\rangle$. The <Cauchy-Schwarz inequality> and <Plancherel theorem> yield the moment product bound $1/4$ with the angular-frequency Fourier convention. Equality occurs exactly for $f(x)=C e^{-ax^2/2}$ with $a>0$ real and $C\ne0$. Translation, modulation or a quadratic phase increase one of these uncentred moments; the equality family is narrower than that for centred variances.
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