A forgotten Theorem of Schoenberg on one-sided integral averages
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Let f:R→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 ≤ t. A natural way to do so is via a weight ϕ and g(t) = ∫_0^∞f(t-s) ϕ(s) ds. We would like that (1) constant functions, f(t) ≡const, are mapped to themselves and (2) ϕ 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 ϕ(s) = λ^-1 e^-λ s for some λ > 0.
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