A constraint on extensible quadrature rules

TL;DR

The paper establishes a fundamental limitation on the growth rate of sample sizes in extensible quadrature rules, showing they must grow at least geometrically to match certain decay rates of integration error.

Contribution

It proves that for certain decay rates of integration error, sample sizes in extensible quadrature rules must grow at least geometrically, ruling out arithmetic progressions.

Findings

Sample sizes must grow at least geometrically for certain error decay rates.

Arithmetic sequences are ruled out as sample size progressions.

Sample size doubling is not ruled out by the established constraint.

Abstract

When the worst case integration error in a family of functions decays as $n^{-\alpha}$ for some $\alpha>1$ 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, $n_{k+1}/n_k\ge \rho$ must hold for a value $1<\rho<2$ that increases with $\alpha$. This result always rules out arithmetic sequences but never rules out sample size doubling. The same constraint holds in a root mean square setting.