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Bernstein's theorem states that for any completely monotone function $f$: $f \in C^{\infty}[0,+\infty)$, $(-1)^n f^{(n)}(t) \geqslant 0$ there is a finite Borel measure $\mu$ such that $$ f(t) = \int_{0}^{+\infty} e^{-tx} \mu(dx) $$

Is there some generalisation of this result on the case of $n$ dimensions?

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Yes, there is and it goes by the name of Bochner theorem. For details see the book of A. Klenke: Probability Theory. A Comprehensive Course, Springer Verlag, 2008.

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    $\begingroup$ Another useful reference is Section 4.2 of Bochner's book "Harmonic analysis and the theory of probability" (which has been reprinted by Dover). $\endgroup$
    – Henry Cohn
    Jan 24, 2012 at 13:11
  • $\begingroup$ That;'s a bit of good news.I did not know it was reprinted. $\endgroup$
    – Liviu Nicolaescu
    Jan 24, 2012 at 13:18
  • $\begingroup$ Thanks, but I rather looked for the next generalisation: is there some similar theorem that characterises functions $f \colon \mathbb{R}^n \to \mathbb{R}$ such that $(-1)^{|\alpha|} \frac{ \partial^{\alpha} }{ \partial^{\alpha_1} \partial^{\alpha_2} \cdots \partial^{\alpha_n} } f(x) \geqslant 0$ ? $\endgroup$
    – Appliqué
    Jan 24, 2012 at 15:46
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    $\begingroup$ See Theorem 4.2.1 in Bochner's book. $\endgroup$
    – Henry Cohn
    Jan 24, 2012 at 15:52
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Further generalizations using the framework of semigroups with involution can be found in the nice book: Harmonic Analysis on Semigroups: theory of positive definite and related functions, by Christian Berg, Jens Peter Reus Christensen, Paul Ressel.

There, the general framework yields the Bernstein-Widder characterization, Bochner's theorem, etc., all as special cases of a more general setup. (The book is quite accessible after Chapter 3).

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