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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



Strict definiteness of integrals via
complete monotonicity of derivatives

Author: L. Mattner
Journal: Trans. Amer. Math. Soc. 349 (1997), 3321-3342
MSC (1991): Primary 26D15, 43A35, 31A15, 60E15
MathSciNet review: 1422615
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Abstract: Let $k$ be a nonnegative integer and let $\varphi : (0,\infty ) \rightarrow \Bbb R$ be a $C^\infty $ function with $(-)^k\cdot \varphi ^{(k)}$ completely monotone and not constant. If $\sigma \neq 0$ is a signed measure on any euclidean space $\Bbb R^d$, with vanishing moments up to order $k-1$, then the integral $ \int _{\Bbb R^d} \int _{\Bbb R^d} \varphi ( \|x-y\|^2 ) \, d\sigma (x) d\sigma (y)$ is strictly positive whenever it exists. For general $d$ no larger class of continuous functions $\varphi $ seems to admit the same conclusion. Examples and applications are indicated. A section on ''bilinear integrability'' might be of independent interest.

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Additional Information

L. Mattner
Affiliation: Universität Hamburg, Institut für Mathematische Stochastik, Bundesstr. 55, D–20146 Hamburg, Germany

Keywords: Bernstein functions, Besicovitch covering theorem, bilinear integrability, conditionally positive definite functions, determinate moment problem, energy integrals, integral inequalities, logarithmic potential theory, moment inequalities, radial analysis
Received by editor(s): January 28, 1996
Dedicated: Dedicated with gratitude to Professor Erwin Mues, on the occasion of his sixtieth birthday
Article copyright: © Copyright 1997 American Mathematical Society