Lemma 32.4.5.label Let $G$ be a locally compact group, then for any $f * g \in C_{c}(G; \complex) * C_{c}(G; \complex)$, there exists $\bracsn{h_k}_{0}^{3} \subset C_{c}(G; \complex) \cap P(G)$ such that
\[f * g = \sum_{k = 0}^{3} i^{k} h_{k}\]
Proof. For each $h \in C_{c}(G; \complex)$, denote $\tilde h(x) = \ol{h(x^{-1})}$, then by polarisation,
\[f * g = \frac{1}{4}\sum_{k = 0}^{3} i^{k} (f + i^{k} \tilde g) * (f + i^{k} \tilde g)^{\sim}\]
Since $C_{c}(G; \complex) \subset L^{2}(G; \complex)$, (2) of Lemma 32.4.4 implies that $(f + i^{k} \tilde g) * (f + i^{k} \tilde g)^{\sim}$ is positive-definite.$\square$
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