Positive-definite function on a group function on a group

Question

I have quite a hard time understanding the definition of positive-definite functions that is based on Hilbert spaces, the one that I read from Wiki; it does not exactly specify that how $H$ relates to $G$? Or if the definition would makes sense for functions with compact support(for locally compact groups $G$) rather than a finite one. Can somebody clarify the formal definition please in this context?

Answer 1

Thanks to CameronWilliams I got the following definition from Folland's Abstract Harmonic Analaysis: {{ A function of positive type on a locally compact group $G$ is a function $\phi\in L^{\infty}(G)$ that defines a positvie linear functional one the Banach *-algebra $L^1(G)$, i.e. that satisfies: $$\forall f\in L^1{G}: \int (f^**f)\phi\geq0$$ We have: $$\int (f^**f)\phi=\int\int\Delta(y^{-1})\bar{f(y^{-1})}f(y^{-1}x)\phi(x)dydx$$

so reversing the order of integration and substiuting $y^{-1}x$ for $x$ shows the $\phi$ is of positivre type if and only if $$\int\int f(x)\bar{f(y)}\phi(y^{-1}x)dydx\geq 0.$$ }}

I can relate to the final equation but I still don't see where $H$ in wiki definition is coming from.