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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 $\mathbb{R}_{\geq 0}$, but how about positive definite function?

Is every positive definite function on $\mathbb{R}_{\geq 0}$ of the form $f(x)=\int_{-\infty}^\infty e^{-tx}\,d\mu(t)$ for some positive measure on $\mathbb{R}_{\geq 0}$?

Here a continuous function $f(x)$ on $\mathbb{R}_{\geq 0}$ is called positive definite if $\sum_{k,l}^Na_k\overline{a_l}f(x_k+x_l)\geq 0$ 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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    $\begingroup$ you have Bochner's theorem at least math.iit.edu/~fass/603_ch2.pdf. Also check out Theorem 2.4.2 and Theorem 2.5.3 if you also have radial. $\endgroup$
    – Thomas Kojar
    Commented Jul 25, 2023 at 0:16
  • $\begingroup$ Thank you for your referece, but the definition of positive definiteness I adopt differs from the one. $\endgroup$
    – user509119
    Commented Jul 25, 2023 at 0:25
  • $\begingroup$ It is not $f(x_k-x_l)$ but $f(x_k+x_l)$. $\endgroup$
    – user509119
    Commented Jul 25, 2023 at 0:34
  • $\begingroup$ So I am taking a look whether you can at least get Bochner back. I am looking at the Folland proof typed here too online individual.utoronto.ca/jordanbell/notes/bochnertheorem.pdf. It seems to me that you still get Theorem 3 because I think you can just skip the very last inequality and you get nonnegative where he has $\phi(x+y)$. $\endgroup$
    – Thomas Kojar
    Commented Jul 25, 2023 at 0:46
  • $\begingroup$ This nonnegative is crucial to Hermitian inner product. $\endgroup$
    – Thomas Kojar
    Commented Jul 25, 2023 at 0:48

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