Positive-definite function on a group

A continuous function $f\neq0$ on the group $G$ such that for all $x_1,\dots,x_n$ in $G$ and $c_1,\dots,c_n\in\mathbf C$,

$$\sum_{i,j}f(x_ix_j^{-1})c_i\overline{c_j}\geq0.$$

Examples can be obtained as follows. Let $\pi\colon G\to\Aut(H)$ be a unitary representation of $G$ in a Hilbert space $H$, and let $u$ be a unit (length) vector. Then

$$f(x)=\langle\pi(x)u,u\rangle$$

is a positive-definite function.

Essentially, these are the only examples. Indeed, there is a bijection between positive-definite functions on $G$ and isomorphism classes of triples $(\pi,H,u)$ consisting of a unitary representation $\pi$ of $G$ on $H$ and a unit vector $u$ that topologically generates $H$ under $\pi(G)$ (a cyclic vector). This is the (generalized) Bochner–Herglotz theorem.

See also Fourier–Stieltjes transform (when $G=\mathbf R$).

References

[a1]	S. Lang, "$\SL_2(\mathbf R)$" , Addison-Wesley (1975) pp. Chap. IV, §5
[a2]	G.W. Mackey, "Unitary group representations in physics, probability and number theory" , Benjamin (1978) pp. 147ff