Reflection positivity on real intervals

Understand

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.