Lemma 6

Tag 0368, permanent. Cite it as [Bourbaki, Tag 0368].

Théories spectrales, chapter IV, § 5. Opérateurs partiels normaux et théorème spectral

Let $\varepsilon >0$ . The function $\sigma_{\varepsilon}$ defined on $\mathbf{R}$ by

\sigma_{\varepsilon}(x) =\frac{2i\varepsilon x}{x^2 + \varepsilon^2}

belongs to $L^2(\mathbf{R})$ . Its Fourier transform is the class in $L^2(\mathbf{R})$ of the function $\eta_{\varepsilon}$ which vanishes at 0 and satisfies

\eta_{\varepsilon}(y) =\frac{2\pi \varepsilon y}{|y|}e^{-2\pi \varepsilon|y|}

for every $y\not = 0$ .

One has $\sigma_{\varepsilon}\in L^2(\mathbf{R})$ by prop. 3 of IV, p. 199, and the class of the function $\eta_{\varepsilon}$ belongs to $L^2(\mathbf{R})\cap L^1(\mathbf{R})$ . For every $x\in \mathbf{R}$ , one has

\overline{\mathscr{F}}(\eta_{\varepsilon})(x) = 2\pi \varepsilon \int_{\mathbf{R}_+}e^{2\pi(ix-\varepsilon)y}dy-2\pi \varepsilon \int_{\mathbf{R}_-}e^{2\pi(ix+\varepsilon)y}dy

=\frac{\varepsilon}{\varepsilon-ix}-\frac{\varepsilon}{\varepsilon + ix}=\frac{2i\varepsilon x}{x^2 + \varepsilon^2}

whence $\overline{\mathscr{F}}(\eta_{\varepsilon}) =\sigma_{\varepsilon}$ . The result then follows from the fourier inversion formula in $L^2(\mathbf{R})$ (corollary of theorem 2 of II, p. 220).

Bourbaki sets the proof directly after the statement, and the corpus does not separate them, so what is above is the statement and its proof together.

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