Lemma 32.4.1.label Let $G$ be a locally compact group, $\phi \in L^{\infty}(G; \complex)$, and $f \in L^{1}(G; \complex)$, then
\[\int_{G} (f^{*} * f) \phi = \iint_{G \times G}f(x)\ol{f(y)}\phi(y^{-1}x)dydx\]
Proof.
\begin{align*}\int_{G} (f^{*} * f)(x)\phi(x) dx&= \iint_{G \times G}\Delta_{G}(y^{-1})\ol{f(y^{-1})}f(y^{-1}x)\phi(x)dydx \\&= \iint_{G \times G}\ol{f(y)}f(yx)\phi(x)dy dx \\&= \iint_{G \times G}f(x) \ol{f(y)}\phi(y^{-1}x)dydx\end{align*}
$\square$
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