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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 $x_1,\cdots,x_N\in\mathbb{R}_{\geq 0}$.

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  • $\begingroup$ This is the same question as in your post. Also, you should not add questions, especially after your original question has been answered. Moreover, Mathoverflow users are asked to avoid answering posts with multiple questions. $\endgroup$
    – Iosif Pinelis
    Commented Jul 25, 2023 at 1:30

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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.

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  • $\begingroup$ Thank your for your example. $\endgroup$
    – user509119
    Commented Jul 25, 2023 at 0:10

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