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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+y2) 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 , ∞) 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-y2) 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.

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Semigroup Forum

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