Bochner and Bernstein theorems via the nuclear integral representation theorem
type: Item
instance of: scholarly article
title: Bochner and Bernstein theorems via the nuclear integral representation theorem
zbMATH Open document ID: 1066.46004
author: Erik G. F. Thomas
published in: Journal of Mathematical Analysis and Applications
publication date: 2004-10-12T00:00:00Z
review text: 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: \(\rho(u) = J_0(2\sqrt{-u}) = I_0(2\sqrt{u})\).
Mathematics Subject Classification ID: 46A55
Mathematics Subject Classification ID: 46A11
Mathematics Subject Classification ID: 46E10
Mathematics Subject Classification ID: 44A10
Mathematics Subject Classification ID: 46F12
zbMATH DE Number: 2107211
zbMATH Keywords: conuclear cones
zbMATH Keywords: completely monotonic functions
zbMATH Keywords: extreme elements
zbMATH Keywords: distributions of positive type
zbMATH Keywords: special functions
reviewed by: Heinrich von Weizsäcker
MaRDI profile type: Publication
full work available at URL: j.jmaa.2004.03.046
OpenAlex ID: W2012657138
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DOI: 10.1016/J.JMAA.2004.03.046
description: scientific article; zbMATH DE number 2107211