We study functions f:(a,b)→R\documentclass[12pt]{minimal}
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\begin{document}$$f : (a,b) \rightarrow {{\mathbb {R}}}$$\end{document} on open intervals in R\documentclass[12pt]{minimal}
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\begin{document}$${{\mathbb {R}}}$$\end{document} with respect to various kinds of positive and negative definiteness conditions. We say that f is positive definite if the kernel f(x+y2)\documentclass[12pt]{minimal}
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\begin{document}$$f\big (\frac{x + y}{2}\big )$$\end{document} is positive definite. We call f negative definite if, for every h>0\documentclass[12pt]{minimal}
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\begin{document}$$h > 0$$\end{document}, the function e-hf\documentclass[12pt]{minimal}
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\begin{document}$$e^{-hf}$$\end{document} 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,∞)\documentclass[12pt]{minimal}
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\begin{document}$$(a,b) = (0,\infty )$$\end{document} it generalizes classical results by Bernstein and Horn. On a symmetric interval (-a,a)\documentclass[12pt]{minimal}
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\begin{document}$$(-a,a)$$\end{document}, we call f reflection positive if it is positive definite and, in addition, the kernel f(x-y2)\documentclass[12pt]{minimal}
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\begin{document}$$f\big (\frac{x - y}{2}\big )$$\end{document} is positive definite. We likewise define reflection negative functions and obtain a Lévy–Khintchine formula for reflection negative functions on all of R\documentclass[12pt]{minimal}
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\begin{document}$${{\mathbb {R}}}$$\end{document}. Finally, we obtain a characterization of germs of reflection negative functions on 0-neighborhoods in R\documentclass[12pt]{minimal}
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\begin{document}$${{\mathbb {R}}}$$\end{document}.
