A constraint on extensible quadrature rules

When the worst case integration error in a family of functions decays as $n^{-α}$ for some $α>1$ and simple averages along an extensible sequence match that rate at a set of sample sizes $n_1<n_2<\dots<\infty$, then these sample sizes must grow at least geometrically. More precisely, $n_{k+1}/n_k\ge ρ$ must hold for a value $1<ρ<2$ that increases with $α$. This result always rules out arithmetic sequences but never rules out sample size doubling. The same constraint holds in a root mean square setting.

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