Remark 1

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

Théories spectrales, chapter II, § 1. Transformation de Fourier

Certain formulas concerning the Fourier transform on $L^1(G)$ extend to the Fourier transform on $L^2(G)$ . In particular, for $f\in L^2(G)$ and $\chi \in \widehat{G}$ , we have

\overline{\mathscr{F}}(f) = (\chi \mapsto \mathscr{F}(f)(\chi^{-1})) =\overline{\mathscr{F}(\overline{f})}

\mathscr{F}(\varepsilon_x*f) =\eta (x^{-1})\mathscr{F}(f),\overline{\mathscr{F}}(\varepsilon_x*f) =\eta (x)\mathscr{F}(f)

\mathscr{F}(\chi f) =\varepsilon_{\chi}*\mathscr{F}(f),\overline{\mathscr{F}}(\chi f) =\varepsilon_{\chi}*\mathscr{F}(f)

If $\sigma$ is an automorphism of G and Δ the module of $\sigma$ (INT, VII, §1, No. $^o4$ , def. 4), then for $f\in L^2(G)$ , one has

\mathscr{F}(f\circ \sigma ) = \Delta^{-1}\mathscr{F}(f)\circ \widehat{\sigma}^{-1}

in $L^2(G)$ .

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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