Semigroup Forum

, Volume 96, Issue 1, pp 31–48 | Cite as

Reflection positivity on real intervals

  • Palle E. T. Jorgensen
  • Karl-Hermann Neeb
  • Gestur ÓlafssonEmail author


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 \({{\mathbb {R}}}\).


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


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Copyright information

© Springer Science+Business Media New York 2017

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

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