Lemma 32.4.4.label Let $G$ be a locally compact group, then:
- (1)
For any unitary representation $(H, \pi)$ of $G$ and $\xi \in H$, $\phi(x) = \dpn{\pi(x)\xi, \xi}{H}$ is a positive-definite function.
- (2)
For any $f \in L^{2}(G; \complex)$, let $\tilde f(x) = \ol{f(x^{-1})}$, then $f * \tilde f$ is a positive-definite function.
Proof. (1): Let $f \in L^{1}(G; \complex)$, then by Lemma 32.4.1,
\begin{align*}\int_{G} (f^{*} * f)(x)\phi(x)dx&= \iint_{G \times G}f(x)\ol{f(y)}\phi(y^{-1}x)dxdy \\&= \iint_{G} \dpn{f(x)\pi(x)\xi, f(y)\pi(y)\xi}{H}dxdy = \norm{\pi(f)\xi}_{H}^{2} \ge 0\end{align*}
(2): Let $\pi: G \to B(L^{2}(G; \complex))$ be the left regular representation of $G$, then
\[\dpn{\pi(x)f, f}{L^2(G; \complex)}= \int_{G} f(x^{-1}y)\ol{f(y)}dy = \ol{(f * \tilde f)(x)}\]
By (1) and Lemma 32.4.3, $f * \tilde f$ is positive-definite.$\square$
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