Semigroup Forum

, Volume 96, Issue 1, pp 31–48

# Reflection positivity on real intervals

• Palle E. T. Jorgensen
• Karl-Hermann Neeb
• Gestur Ólafsson
RESEARCH ARTICLE

## Abstract

We study functions $$f : (a,b) \rightarrow {{\mathbb {R}}}$$ on open intervals in $${{\mathbb {R}}}$$ with respect to various kinds of positive and negative definiteness conditions. We say that f is positive definite if the kernel $$f\big (\frac{x + y}{2}\big )$$ is positive definite. We call f negative definite if, for every $$h > 0$$, the function $$e^{-hf}$$ is positive definite. Our first main result is a Lévy–Khintchine formula (an integral representation) for negative definite functions on arbitrary intervals. For $$(a,b) = (0,\infty )$$ it generalizes classical results by Bernstein and Horn. On a symmetric interval $$(-a,a)$$, we call f reflection positive if it is positive definite and, in addition, the kernel $$f\big (\frac{x - y}{2}\big )$$ is positive definite. We likewise define reflection negative functions and obtain a Lévy–Khintchine formula for reflection negative functions on all of $${{\mathbb {R}}}$$. Finally, we obtain a characterization of germs of reflection negative functions on 0-neighborhoods in $${{\mathbb {R}}}$$.

## Keywords

Positive definite function Negative definite function Bernstein function Reflection positive function Reflection negative function

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## Authors and Affiliations

• Palle E. T. Jorgensen
• 1
• Karl-Hermann Neeb
• 2
• Gestur Ólafsson
• 3
Email author
1. 1.Department of MathematicsThe University of IowaIowa CityUSA
2. 2.Department MathematikFAU Erlangen-NürnbergErlangenGermany
3. 3.Department of MathematicsLouisiana State UniversityBaton RougeUSA