Bochner and Bernstein theorems via the nuclear integral representation theorem (Q1883362)

The nuclear integral representation theorem mentioned in the title is a variant due to the present author [J. Convex Anal. 1, 225--258 (1994; Zbl 0837.46009)] of the classical Choquet representation theorem which works in conuclear cones. In this paper, the author reviews this result and applies it to give new proofs of (a) the Bochner-Schwartz characterization of distributions of positive type and (b) the Bernstein characterization of Laplace transformations in finite-dimensional proper cones. This includes the corresponding characterizations of extremal elements. New is the following special case: For \(\alpha > 0\), the function \(f_\alpha(x,y) = 1/(1+x+y+\alpha xy)\) on the positive quadrant is completely monotonic if and only if \(\alpha \in (0,1]\). The corresponding measure is given by \[ \nu_\alpha(ds,d\tau) = \frac{1}{\alpha}e^{-s/\alpha}e^{-\tau/\alpha} \rho\big((1-\alpha)(s/\alpha)(\tau/\alpha)\big)dsd\tau \] where \(\rho\) is essentially a Bessel function: \(\rho(u) = J_0(2\sqrt{-u}) = I_0(2\sqrt{u})\).