theorem is as a one-to-one correspondence between continuous positive definite functions and probability measures. This form of the statement is in agreement with Theorem \[thm:bochner\] as long as one requires $f(0) = 1$ for the positive definite functions.

Let $f$ be a positive definite function. If $f^r$ is positive definite for any real power $r \geq 0$, then $f$ is said to be *infinitely divisible*. By definition, any root $f^{1/n}$ of an infinitely divisible function $f$ is a positive definite function. It follows from Theorem \[thm:schur\] that any product of infinitely definite functions is again infinitely divisible. Moreover, $f$ is infinitely divisible if and only if $f^{1/n}$ is positive definite for any $n \in {\mathbb N}$, since products and pointwise limits of positive definite functions are positive definite.

We next recall Schoenberg’s theorem which links completely monotone functions and positive definite functions.

[@MR1503439]\[thm:characterisation\] A function $g \colon [0,\infty)\to {\mathbb R}$ continuous at zero is completely monotone if and only if $g(|\cdot|^2)$ is positive definite on ${\mathbb R}^d$ for all $d \in {\mathbb N}$.

We have the following two results, giving us properties of transforms of completely monotone and Stieltjes functions, respectively.

\[prop:completely monotone Fourier\] Let $f$ and $g$ be two 