View source for Positive-definite function on a group

{{TEX|done}}
A continuous [[Function|function]] $f\neq0$ on the [[Group|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|unitary representation]] of $G$ in a [[Hilbert space|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|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|cyclic vector]]). This is the (generalized) Bochner–Herglotz theorem.

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

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