A forgotten Theorem of Schoenberg on one-sided integral averages

Abstract

Let $f:\mathbb{R} \rightarrow \mathbb{R}$ be a function for which we want to take local averages. Assuming we cannot look into the future, the 'average' at time $t$ can only use $f(s)$ for $s \leq t$. A natural way to do so is via a weight $\phi$ and $$ g(t) = \int_{0}^{\infty}{f(t-s) \phi(s) ds}.$$ We would like that (1) constant functions, $f(t) \equiv \mbox{const}$, are mapped to themselves and (2) $\phi$ to be monotonically decreasing (the more recent past should weigh more heavily than the distant past). Moreover, we want that (3) if $f(t)$ crosses a certain threshold $n$ times, then $g(t)$ should not cross the same threshold more than $n$ times (if $f(t)$ is the outside wind speed and crosses the Tornado threshold at two points in time, we would like the averaged wind speed to cross the Tornado threshold at most twice). A Theorem implicit in the work of Schonberg is that these three conditions characterize a unique weight that is given by the exponential distribution $$ \phi(s) = \lambda^{} e^{-\lambda s} \qquad \mbox{for some} \quad \lambda > 0.$$