Journal article
Reflection positivity on real intervals
Semigroup Forum, Vol.96(1), pp.31-48
02/2018
DOI: 10.1007/s00233-017-9847-8
Abstract
We study functions $$f : (a,b) \rightarrow {{\mathbb {R}}}$$ f:(a,b)→R on open intervals in $${{\mathbb {R}}}$$ R with respect to various kinds of positive and negative definiteness conditions. We say that f is positive definite if the kernel $$f\big (\frac{x + y}{2}\big )$$ f(x+y2) is positive definite. We call f negative definite if, for every $$h > 0$$ h>0 , the function $$e^{-hf}$$ 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 $$(a,b) = (0,\infty )$$ (a,b)=(0,∞) it generalizes classical results by Bernstein and Horn. On a symmetric interval $$(-a,a)$$ (-a,a) , we call f reflection positive if it is positive definite and, in addition, the kernel $$f\big (\frac{x - y}{2}\big )$$ f(x-y2) is positive definite. We likewise define reflection negative functions and obtain a Lévy–Khintchine formula for reflection negative functions on all of $${{\mathbb {R}}}$$ R . Finally, we obtain a characterization of germs of reflection negative functions on 0-neighborhoods in $${{\mathbb {R}}}$$ R .
Details
- Title: Subtitle
- Reflection positivity on real intervals
- Creators
- Palle Jorgensen - 0000 0004 1936 8294 grid.214572.7 Department of Mathematics The University of Iowa Iowa City IA 52242 USAKarl-Hermann Neeb - 0000 0001 2107 3311 grid.5330.5 Department Mathematik FAU Erlangen-Nürnberg Cauerstrasse 11 91058 Erlangen GermanyGestur Ólafsson - 0000 0001 0662 7451 grid.64337.35 Department of Mathematics Louisiana State University Baton Rouge LA 70803 USA
- Resource Type
- Journal article
- Publication Details
- Semigroup Forum, Vol.96(1), pp.31-48
- Publisher
- Springer US; New York
- DOI
- 10.1007/s00233-017-9847-8
- ISSN
- 0037-1912
- eISSN
- 1432-2137
- Language
- English
- Date published
- 02/2018
- Academic Unit
- Mathematics
- Record Identifier
- 9983985950402771
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