Bochner's theorem

Remarks

The easy direction: if $\varphi$ has this form, then $\sum_{j,k} c_j \overline{c_k} \varphi(x_j - x_k) = \int_{\widehat{G}} \abs{\sum_j c_j \chi(x_j)}^2 \, d\mu(\chi) \ge 0$. For the converse, $\varphi$ defines a positive linear functional on $L^1(G)$, hence on the commutative group C*-algebra of $G$. The spectrum of this algebra is $\widehat{G}$, and the Riesz representation theorem then gives $\mu$. Example: for $G = \R$, the function $\varphi(x) = e^{-x^2/2}$ is positive-definite, and $\mu$ is the standard normal distribution, since $\int_\R e^{i\xi x} \frac{1}{\sqrt{2\pi}} e^{-\xi^2/2} \, d\xi = e^{-x^2/2}$. For $G = \R$, the theorem says that the continuous positive-definite functions with $\varphi(0) = 1$ are exactly the characteristic functions of probability distributions. Continuity is needed: on $\R$ the indicator function of $\set{0}$ is positive-definite but is not the transform of any measure, since such transforms are continuous.

Going deeper

For a student

The theorem is a key tool in the proof of the Plancherel theorem for abelian groups and of Pontryagin duality. For $G = \Z$ it is Herglotz's theorem on positive-definite sequences, with measures on the circle. In the theory of stationary processes it gives the spectral measure of a stationary covariance function.