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://dx.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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