Uncentred Fourier uncertainty principle ($\Vert xf{\Vert }_2\Vert f^{\prime }{\Vert }_2\ge \Vert f{\Vert }_2^2/2$)

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