By Hausdorff-Bernstein-Widder theorem, any completely monotonic function on the half line $\mathbb{R}\_{\geq 0}:=[0,\infty)$ is given by the Laplace transform of a positive measure on $\mathbb{R}\_{\geq 0}$.


My question is following:


Is there a positive definite function on $\mathbb{R}\_{\geq 0}$ which is not given by the Laplace transform of a positive measure on $\mathbb{R}\_{\geq 0}$.


Here a continuous function $f(x)$ on $\mathbb{R}\_{\geq 0}$ is called positive definite if
$\sum\_{k,l=1}^Na\_k\overline{a\_l}f(x\_k+x\_l)\geq 0$ is satisfied for all $a\_1,\cdots,a\_N \in\mathbb{C}$ and $x\_1,\cdots,x\_N\in\mathbb{R}\_{\geq 0}$.



Positive definite but not completely monotonic function on the upper half line
Counterexample: If $f(x)=e^x$ for real $x\ge0$, then $f$ is positive definite. However, $f$ is not the Laplace transform of a positive measure on $[0,\infty)$ -- because otherwise $f$ would be nonincreasing.


