Reflection positivity on real intervals
Jorgensen PE, Neeb KH, Olafsson G (2018)
Publication Type: Journal article
Publication year: 2018
Journal
Book Volume: 96
Pages Range: 31-48
DOI: 10.1007/s00233-017-9847-8
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 L\'evy--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 L\'evy--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.
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APA:
Jorgensen, P.E., Neeb, K.H., & Olafsson, G. (2018). Reflection positivity on real intervals. Semigroup Forum, 96, 31-48. https://doi.org/10.1007/s00233-017-9847-8
MLA:
Jorgensen, Palle E.T., Karl Hermann Neeb, and Gestur Olafsson. "Reflection positivity on real intervals." Semigroup Forum 96 (2018): 31-48.
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