The Best Way to Reconcile Your Past Is Exponentially

Abstract

Let $f:\mathbb{R}\to \mathbb{R}$ be a continuous function for which we want to take local averages. Assuming we cannot look into the future, a commonly encountered problem is that the “average” g(t) at time t can only use f(s) for $s\le t$. A natural way to compute the average is via a weighting function $\varphi :[0,\infty ]\to \mathbb{R}$ and define an average as an integral over $f(t-s)$ weighted by $\varphi (s)$. We would like (1) constant functions, $f(t)\equiv \text{const}$, to be mapped to themselves, and (2) $\varphi$ to be monotonically decreasing (the more recent past should weigh more heavily than the distant past). Moreover, (3) if f(t) crosses a certain threshold n times, then g(t) should not cross the same threshold more than n times. A theorem implicit in the work of Schönberg is that these three conditions characterize a unique weight and $\varphi (s)={\lambda }^{}{e}^{-\lambda s}$ for some $\lambda >0.$

• MSC: Primary 44-03
• Secondary 44A35

Acknowledgments

The author is supported by the NSF (DMS-1763179) and the Alfred P. Sloan Foundation.