Reflection positivity on real intervals

We study functions \(f : (a,b) \rightarrow {{\mathbb {R}}}\) on open intervals in \({{\mathbb {R}}}\) with respect to various kinds of positive and negative definiteness conditions. We say that f is positive definite if the kernel \(f\big (\frac{x + y}{2}\big )\) 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 \(f\big (\frac{x - y}{2}\big )\) is positive definite. We likewise define reflection negative functions and obtain a Lévy–Khintchine formula for reflection negative functions on all of \({{\mathbb {R}}}\). Finally, we obtain a characterization of germs of reflection negative functions on 0-neighborhoods in \({{\mathbb {R}}}\).