Exercise 27, § 1

Tag 02L3, permanent. Cite it as [Bourbaki, Tag 02L3].

Spectral Theories, II, p. 273

Let $f$ be a function in $\mathscr{S}(\mathbf{R})$ of norm $L^2$ equal to 1. One has (uncertainty principle) the inequality

(\int x^2|f(x)|^2dx)(\int t^2|\widehat{f}(t)|^2dt)\geqslant 1_2

_{\mathbf{R}}\mathbf{R}(4\pi )

(note that

-\frac{1}{2}=\mathscr{R}(\int_{\mathbf{R}}xf(x)\overline{f'(x)}dx)

and apply the Cauchy-Schwarz inequality). Determine the cases of equality.

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