Reflection positivity on real intervals
Abstract
We study functions f : (a,b) ---> R on open intervals in R with respect to various kinds of positive and negative definiteness conditions. We say that f is positive definite if the kernel f((x + y)/2) 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((x - y)/2) is positive definite. We likewise define reflection negative functions and obtain a Lévy--Khintchine formula for reflection negative functions on all of R. Finally, we obtain a characterization of germs of reflection negative functions on 0-neighborhoods in R.
- Publication:
-
arXiv e-prints
- Pub Date:
- August 2016
- DOI:
- 10.48550/arXiv.1608.04010
- arXiv:
- arXiv:1608.04010
- Bibcode:
- 2016arXiv160804010J
- Keywords:
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- Mathematics - Functional Analysis;
- 43A35
- E-Print:
- Final version, to appear in "Semigroup Forum"